Flops and spherical functors

Abstract

We study derived categories of Gorenstein varieties $X$ and $X^+$ connected by a flop. We assume that the flopping contractions $f\colon X\to Y$, $f^+ \colon X^+ \to Y$ have fibers of dimension bounded by one and $Y$ has canonical hypersurface singularities of multiplicity two. We consider the fiber product $W=X\times _YX^+$ with projections $p\colon W\to X$, $p^+\colon W\to X^+$ and prove that the flop functors $F = Rp^+_*Lp^* \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X^+)$, $F^+= Rp_*L{p^+}^* \colon {\mathcal {D}}^b(X^+) \to {\mathcal {D}}^b(X)$ are equivalences, inverse to those constructed by Van den Bergh. The composite $F^+ \circ F \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X)$ is a non-trivial auto-equivalence. When variety $Y$ is affine, we present $F^+ \circ F$ as the spherical cotwist of a spherical couple $(\Psi ^*,\Psi )$ which involves a spherical functor $\Psi$ constructed by deriving the inclusion of the null category $\mathscr {A}_f$ of sheaves ${\mathcal {F}} \in \mathop {{\rm Coh}}\nolimits (X)$ with $Rf_*({\mathcal {F}} )=0$ into $\mathop {{\rm Coh}}\nolimits (X)$. We construct a spherical pair (${\mathcal {D}}^b(X)$, ${\mathcal {D}}^b(X^+)$) in the quotient ${\mathcal {D}}^b(W) /{\mathcal {K}}^b$, where ${\mathcal {K}}^b$ is the common kernel of the derived push-forwards for the projections to $X$ and $X^+$, thus implementing in geometric terms a schober for the flop. A technical innovation of the paper is the $L^1f^*f_*$ vanishing for Van den Bergh's projective generator. We construct a projective generator in the null category and prove that its endomorphism algebra is the contraction algebra.

1. Introduction

A homological interpretation of the minimal model program (MMP) in birational geometry was proposed in [BO95, BO02]. The basic idea is that MMP is about ‘minimisation’ of the derived category ${\mathcal {D}}^b(X)$ of coherent sheaves on a variety $X$ for varieties in a given birational class. More precisely, it is expected that if $X$ allows a divisorial contraction or a flip $X\dashrightarrow Y$, then ${\mathcal {D}}^b(X)$ has a semi-orthogonal decomposition (SOD) with one component of SOD equivalent to ${\mathcal {D}}^b(Y)$. Thus, minimizing the birational model should have the categorical meaning of chopping off semi-orthogonal factors of the derived category. A minimal model is expected to be a representative in the birational class of a variety whose derived category does not allow semi-orthogonal factors equivalent to the derived category of a variety birationally equivalent to $X$.

As MMP in dimension greater than two deals with singular varieties, the right choice of the derived category to consider also matters. In particular, for ${\mathbb {Q}}$-Gorenstein varieties the derived category of a suitable stack is relevant (cf. [Kaw02, Kaw09]). Minimal models are not unique, and MMP considers birational maps, called flops, that link various minimal models. Conjecturally, flops induce derived equivalences [BO95, Kaw02, BO02].

There is substantial evidence in favour of this conjectural picture, starting from the original paper [BO95], where various instances of flops were proved to induce derived equivalences, and for simple higher-dimensional flips, the required SOD was constructed.

We expect that the whole zoology of categories, functors and natural transformations relevant to MMP should be governed by interesting hidden homotopy types, maps and higher homotopies. The present work can be considered as taking steps in this direction for categories and functors invoked by flops.

It was mentioned by the authors of [BO95] that the functor that provides an equivalence ${\mathcal {D}}^b(X)\to {\mathcal {D}}^b(X^+)$, where $X$ and $X^+$ are connected by a flop, when composed with the analogous functor in the opposite direction ${\mathcal {D}}^b(X^+)\to {\mathcal {D}}^b(X)$ is not the identity but produces a non-trivial auto-equivalence of ${\mathcal {D}}^b(X)$ (nowadays called a flop–flop functor). For Atiyah flop, the functor is given by what is now known as the spherical twist with respect to the spherical object ${\mathcal {O}}_C(-1)$, where $C$ is the (rational) exceptional curve. It was probably one of the first appearances of the spherical twists. (However, we should also mention here the work of Mukai [Muk87] and Kuleshov [Kul90] who used the action of spherical twists in their non-derived version to describe moduli of sheaves on K3 surfaces. A quick generalisation to the Calabi–Yau/derived case was understood by the authors of [BO95] in those days of sturm und drang on derived categories in Moscow in the early 1990s). Kontsevich suggested that the spherical twist of Atiyah flop should be transferred by mirror symmetry into the equivalence of the Fukaya category induced by the Dehn twist along a vanishing cycle. Properties of spherical twists with respect to spherical objects were later scrutinised by Seidel and Thomas in [ST01] and for more general spherical functors by Anno and Logvinenko in [AL17].

Spherical (co)twist is a unification tool for various non-trivial auto-equivalences of ${\mathcal {D}}^b(X)$ (cf. [Add16]) such as tensor products with line bundles, twists around spherical objects [ST01], EZ-twists [Hor05] or window shifts [DS14].

The homotopy meaning of spherical (co)twists can be read off from their interpretation via schobers, i.e. categorifications of perverse sheaves on stratified topological spaces, suggested by Kapranov and Schechtman [KS14]. The homotopy type of the underlying stratification (the punctured disc for the case of one spherical functor) encodes the algebra of functors and natural transformations in the schober. The schober on the punctured disc has one of possible incarnations via a spherical pair, i.e. a pair of admissible subcategories in a triangulated category satisfying conditions that imply a spherical functor between them.

We study functors and natural transformations for flops with dimension of fibers of the flopping contractions bounded by one. We construct the spherical pair $({\mathcal {D}}^b(X), {\mathcal {D}}^b(X^+))$ in the appropriate quotient of the derived category of $X\times _Y X^+$. The orthogonal complement to ${\mathcal {D}}^b(X)$ in that quotient is the bounded derived category of the abelian null category

(1)\begin{equation} \mathscr{A}_f= \{E \in \mathop{{\rm Coh}}\nolimits(X)\,|\, Rf_* E =0 \}. \end{equation}

By deriving the embedding ${\mathcal {A}}_f\to \mathop {{\rm Coh}}\nolimits (X)$ we obtain the spherical functor $\Psi \colon {\mathcal {D}}^b(\mathscr {A}_f)\to {\mathcal {D}}^b(X)$. The spherical cotwist of the spherical couple (see Appendix C) $(\Psi ^* ,\Psi )$ is the flop–flop functor.

We lift our functors and natural transformations to bicategories $\mathop {\textbf {Bimod}}\nolimits$ and $\mathop {\textbf {FM}}\nolimits$ (see Appendix C). We systematically consider 2-categorical adjunctions instead of the usual adjunctions of functors, scrutinise the uniqueness of adjoints and of associated twists and cotwists, i.e. the cones of adjunction units and counits.

Now we describe our results in more detail.

The flop functor and Van den Bergh's functor

We consider a flopping contraction $f:X \to Y$ and its flop $f^+: X^+\to Y$ with dimension of fibers bounded by one. Exceptional loci of $f$ and $f^+$ are assumed to have codimension greater than one in $X$, respectively in $X^+$, whereas varieties $X$, $X^+$ and $Y$ are assumed to be Gorenstein and $Y$ is assumed to have canonical hypersurface singularities of multiplicity two.

Consider the diagram for the fiber product:

where $p$ and $p^+$ denote the projections to the factors. The flop functor is

\[ F = Rp^+_*Lp^*\colon{\mathcal{D}}_{qc}(X)\to {\mathcal{D}}_{qc}(X^+). \]

Note that this functor does not necessarily induce an equivalence if the dimension of fibers of $f$ is greater than one (see [Nam04]), which means that the functor needs an adjustment for fibers of higher dimension.

The above assumptions on the flopping contraction are those adopted by Van den Bergh in [VdB04]. We use also some techniques borrowed from his work. Van den Bergh constructed an equivalence ${\Sigma } \colon {\mathcal {D}}^b(X)\xrightarrow {\simeq } {\mathcal {D}}^b(X^+)$, which is given via the identification of both categories with the derived category of modules over a sheaf of non-commutative algebras on affine $Y$. We first give a new interpretation for functor ${\Sigma }$. To this end, we identify ${\mathcal {D}}^b(X)$ with $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})$, the homotopy category of complexes of projective objects $\mathscr {P}_{-1}$ in the heart $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ of the perverse $t$-structure, introduced by Bridgeland in [Bri02]. We show that ${\Sigma }$ can be defined as the functor $(f^{+*}f_*(-))^{\vee \vee }$ applied term-wise to complexes of objects in $\mathscr {P}_{-1}$. We give an independent proof that the flop functor is an equivalence and that it can be extended to an equivalence $\Sigma _{{\rm qc}}$ between unbounded categories of quasi-coherent sheaves (Proposition 4.6).

Then we show that the flop functor is given in the same way by the term-wise application of the functor $f^{+*}f_*(-)$. To this end, we need one of the technical innovations of our paper, the $L^1f^*f_*$-vanishing for objects in $\mathscr {P}_{-1}$. We prove (Lemma 3.4) that if $f \colon X \to Y$ is a flopping contraction satisfying the above conditions, then $L^1f^*f_* {\mathcal {M}}$ is zero, for any ${\mathcal {M}} \in \mathscr {P}_{-1}$. The proof of the vanishing is based on a local presentation of $Y$ as a divisor in a smooth variety ${\mathcal {Y}}$. Note that, since $Y$ is singular, object $Lf^*f_* {\mathcal {M}}$ in general has infinitely many non-zero cohomology sheaves, more precisely, they satisfy 2-periodicity (see § 3.5), which is reminiscent of matrix factorisations [Buc86, Orl04].

We consider a divisorial embedding $i \colon Y \to {\mathcal {Y}}$ into a smooth ${\mathcal {Y}}$ together with $g=i \circ f$, $g^+ = i \circ f^+$. If the base $Y$ of the flopping contraction is affine, then the flop functor $F$ and Van den Bergh's equivalence $\Sigma _{{\rm qc}}\colon {\mathcal {D}}_{{\rm qc}}(X)\to {\mathcal {D}}_{{\rm qc}}(X^+)$ fit into a functorial exact triangle (Proposition 4.13):

(2)\begin{equation} \Sigma_{{\rm qc}}[1] \to Lg^{+*}Rg_* \to F \to \Sigma_{{\rm qc}}[2]. \end{equation}

As both $Lg^{+*}Rg_*$ and $\Sigma _{{\rm qc}}$ take ${\mathcal {D}}^b(X)$ to ${\mathcal {D}}^b(X^+)$, this allows us to conclude that the flop functor also preserves the boundedness of the derived categories.

Theorem 4.21 states that $\Sigma _{{\rm qc}}$ is actually the inverse of the opposite flop functor

\[ F^+ = Rp_*Lp^{+*} \colon {\mathcal{D}}_{{\rm qc}}(X^+) \to {\mathcal{D}}_{{\rm qc}}(X). \]

This implies that flop functors $F$ and $F^+$ yield (not mutually inverse) equivalences between ${\mathcal {D}}^b(X)$ and ${\mathcal {D}}^b(X^+)$. Following an argument of Chen [Che02], we generalise this statement to the case when base $Y$ is quasi-projective (see § 4.6).

In addition, we show that the composite $F^+ F$, the ‘flop–flop’ functor, is the term-wise extension of $f^*f_*(-)|_{\mathscr {P}_{-1}}$.

The null category and spherical functor $\Psi$

One of the original motivations for our work was to recover the importance of the null category for $f$ as in (1). Category $\mathscr {A}_f$ admits a projective generator ${\mathcal {P}}$, which we obtain from Van den Bergh's projective generator $\mathcal {M}$ for the perverse heart $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ by a projection to the null category (see Proposition 2.4). Deriving the embedding $\mathscr {A}_f \to \mathop {{\rm Coh}}\nolimits (X)$ gives us functor

\[ \Psi \colon {\mathcal{D}}^b(\mathscr{A}_f) \to {\mathcal{D}}^b(X). \]

We prove that, for a flopping contraction $f \colon X \to Y$ with affine $Y$, functor $\Psi$ is spherical. The flop–flop functor $F^+F$ is the spherical cotwist of the spherical couple $(\Psi ^*,\Psi )$, i.e. the $\Psi ^* \dashv \Psi$ adjunction unit fits into a functorial exact triangle:

\[ F^+F \to \mathop{{\rm Id}}\nolimits_{{\mathcal{D}}^b(X)} \to \Psi \Psi^* \to F^+F[1]. \]

We also show that the spherical twist ${\mathcal {D}}^b(\mathscr {A}_f) \to {\mathcal {D}}^b(\mathscr {A}_f)$ of the couple $(\Psi ^*,\Psi )$ is the shifted identity functor $\mathop {{\rm Id}}\nolimits _{{\mathcal {D}}^b(\mathscr {A}_f)}[4]$ (see Corollary 5.18).

Spherical pairs

We assume again $f \colon X\to Y$ to be a flopping contraction with affine $Y$. Functor $p^*$ induces an isomorphism of endomorphism algebras of a projective generator ${\mathcal {P}}\in \mathscr {A}_f$ and of $p^* {\mathcal {P}}\in \mathop {{\rm Coh}}\nolimits (X \times _Y X^+)$. Moreover, $\mathop {{\rm Ext}}\nolimits ^i_{X\times _Y X^+}(p^*{\mathcal {P}}, p^* {\mathcal {P}}) = 0$, for $i>0$ (see Proposition 5.3). Thus, we construct a fully faithful functor ${\mathcal {D}}(\mathscr {A}_f) \to {\mathcal {D}}_{{\rm qc}}(X\times _Y X^+)$.

We consider the ‘common kernel subcategories’:

\begin{align*} {\mathcal{K}}^b &= \{ E\in {\mathcal{D}}^b(X\times_Y X^+)\,|\, Rp_*(E) =0,\, Rp^+_*(E) =0\},\\ {\mathcal{K}}^- &= \{ E\in {\mathcal{D}}^-(X\times_Y X^+)\,|\, Rp_*(E) =0,\, Rp^+_*(E) =0\}. \end{align*}

The composite ${\mathcal {D}}^b(X) \xrightarrow {Lp^*} {\mathcal {D}}^-(X\times _Y X^+) \to {\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}^-$ (not $Lp^*$ itself) factors via ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$, thus inducing a fully faithful functor (Proposition 5.8)

\[ \widetilde{L}p^* \colon {\mathcal{D}}^b(X) \to {\mathcal{D}}^b(X\times_Y X^+)/{\mathcal{K}}^b. \]

We prove the existence of SODs (see [Bon89]):

(3)\begin{equation} {\mathcal{D}}^b(X\times_Y X^+)/{\mathcal{K}}^b = \langle {\mathcal{D}}^b(\mathscr{A}_{f^+}),\quad \widetilde{L}p^* {\mathcal{D}}^b(X)\rangle = \langle \widetilde{L}p^* {\mathcal{D}}^b(X), {\mathcal{D}}^b(\mathscr{A}_f)\rangle. \end{equation}

As a result, we obtain a geometric description of the category ${\mathcal {D}}^b(\mathscr {A}_{f^+})$:

\[ {\mathcal{D}}^b(\mathscr{A}_{f^+}) \simeq \{E \in {\mathcal{D}}^b(X\times_Y X^+)\,|\, Rp_*(E) = 0\}/{\mathcal{K}}^b. \]

By exchanging the roles of $X$ and $X^+$ in (3) we obtain SODs

(4)\begin{equation} {\mathcal{D}}^b(X\times_Y X^+)/{\mathcal{K}}^b = \langle {\mathcal{D}}^b(\mathscr{A}_{f}),\quad \widetilde{L}{p^+}^* {\mathcal{D}}^b(X^+)\rangle = \langle \widetilde{L}{p^+}^* {\mathcal{D}}^b(X^+), {\mathcal{D}}^b(\mathscr{A}_{f^+})\rangle. \end{equation}

Together decompositions (3) and (4) provide us with a geometric description of 4-periodical SODs (see Proposition B.3), whose relation to spherical functors was basically discovered by Halpern-Leistner and Shipman [HS16] (we thank Kapranov for explanations on this). Two pairs of subcategories $({\mathcal {D}}^b(X), {\mathcal {D}}^b(X^+))$, $({\mathcal {D}}^b(\mathscr {A}_f), {\mathcal {D}}^b(\mathscr {A}_{f^+}))$ are spherical pairs [KS14] (see Theorem 5.17). The corresponding spherical functor for the second spherical pair is $\Psi$.

The contraction algebra

Assume the base $Y$ of the flopping contraction $f$ to be the spectrum of a complete Noetherian local ring. Then the reduced fiber of $f$ over the unique closed point of $Y$ is a union of $n$ smooth irreducible rational curves $C_1, \ldots, C_n$ (see Theorem D.1). The category $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ has $n+1$ irreducible projective objects ${\mathcal {M}}_0, \ldots, {\mathcal {M}}_n$, with ${\mathcal {M}}_0 \simeq {\mathcal {O}}_X$ (see [VdB04]). ${\mathcal {M}} = \bigoplus _{i=0}^n {\mathcal {M}}_i$ is a projective generator for $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$.

We prove that the endomorphism algebra

\[ A_P = \mathop{{\rm Hom}}\nolimits_{\mathscr{A}_f}({\mathcal{P}}, {\mathcal{P}}) \]

of the corresponding projective generator for $\mathscr {A}_f$, ${\mathcal {P}}= \ker (f^*f_* {\mathcal {M}} \to {\mathcal {M}})$, is isomorphic to the contraction algebra introduced in [IW14], which is defined as the quotient of $\mathop {{\rm Hom}}\nolimits _X( {\mathcal {M}}, {\mathcal {M}})$ by the ideal of morphisms that factor via direct sums of copies of ${\mathcal {O}}_X$ (Theorem 6.2). This theorem relates our work to the results of Donovan and Wemyss [DW16, DW19], where contraction algebras appear in the context of non-commutative deformations.

Appendices

There are five appendices attached to the main body of the paper.

Appendix A. This appendix is an extract of some properties of the functor $f^!$, right dual to $Rf_*$, and the Grothendieck duality.

Appendix B. In this appendix we introduce functorial exact triangles and use them to define spherical functors. We recall after [KS14] the notion of a spherical pair and the associated spherical functor. 4-periodical SODs introduced by Halpern-Leistner and Shipman [HS16] produce spherical pairs.

Appendix C. In this appendix we discuss a bicategory ${\mathcal {C}}$ and a pair of 1-morphisms $s\in \mathop {{\rm Hom}}\nolimits _{{\mathcal {C}}}(A,B)$, $r\in \mathop {{\rm Hom}}\nolimits _{{\mathcal {C}}}(B,A)$ that fit into a 2-categorical adjunction $(s,r,\eta,\varepsilon )$. When ${\mathcal {C}}$ is 1-triangulated (meaning the categories of 1-morphisms are triangulated), we define the twist $t_s\in \mathop {{\rm Hom}}\nolimits _{{\mathcal {C}}}(B,B)$ and the cotwist $c_s\in \mathop {{\rm Hom}}\nolimits _{{\mathcal {C}}}(A,A)$ as the cones of the counit $\varepsilon \colon sr\to \mathop {{\rm Id}}\nolimits _B$ and the unit $\eta \colon \mathop {{\rm Id}}\nolimits _A\to rs$. By using pseudo-functors and 2-categorical equivalences we show that the twist and cotwist are in a suitable sense invariant under replacing $A$ and $B$ by equivalent objects. If $t_s$ and $c_s$ are invertible in the 2-categorical sense, we say that $(s,r,\eta, \varepsilon )$ is a spherical couple.

We lift exact functors between triangulated categories to 1-morphisms in appropriate 1-triangulated bicategories. The first bicategory is $\mathop {\textbf {Bimod}}\nolimits$ whose objects are DG algebras and categories of 1-morphisms are defined as the derived categories of DG bimodules. The second one is the Fourier–Mukai bicategory FM of schemes and the derived categories of coherent sheaves on their products as categories of 1-morphisms. Both bicategories admit 2-functors to the bicategory $\textbf {Cat}$ of categories, functors and natural transformations. More precisely, we have $\Phi \colon \mathop {\textbf {Bimod}}\nolimits \to \textbf {Cat}$, $\Phi (A) = {\mathcal {D}}(A)$ and $\Xi \colon \mathop {\textbf {FM}}\nolimits \to \textbf {Cat}$, $\Xi (X) = {\mathcal {D}}_{{\rm qc}}(X)$.

The above theory of 2-categorical adjunctions ensures that once a lift of an exact functor ${\mathcal {D}}_{{\rm qc}}(X) \to {\mathcal {D}}_{{\rm qc}}(Y)$ to a 1-morphism in $\mathop {\textbf {Bimod}}\nolimits$ or $\mathop {\textbf {FM}}\nolimits$ admitting an adjoint is fixed, we obtain essentially unique exact functors corresponding to the twist and the cotwist.

We describe 2-categorical adjunctions in $\mathop {\textbf {Bimod}}\nolimits$ using formulae for adjoint bimodules as in [AL21]. For ${\mathcal {E}}\in {\mathcal {D}}_{{\rm qc}}(X\times Y)$ we discuss functors adjoint to $\Xi _{\mathcal {E}}\colon {\mathcal {D}}_{{\rm qc}}(X) \to {\mathcal {D}}_{{\rm qc}}(Y)$ and the conditions under which they are FM functors. Results of [LS16] allow us to transfer between $\mathop {\textbf {Bimod}}\nolimits$ and $\mathop {\textbf {FM}}\nolimits$ via fixing compact generators for ${\mathcal {D}}_{{\rm qc}}(X)$ and ${\mathcal {D}}_{{\rm qc}}(Y)$. We use the 2-categorical adjunctions in $\mathop {\textbf {Bimod}}\nolimits$ to construct functorial exact triangles for $\Xi _{\mathcal {E}}$ and its adjoints. We check that these triangles are, up to isomorphism, independent of the choice of compact generators. In particular, given a morphism $f\colon X \to Y$, we discuss the lift of $Rf_*$, its adjoints, and the adjunction (co)units to $\mathop {\textbf {Bimod}}\nolimits$ and $\mathop {\textbf {FM}}\nolimits$. We also describe a 2-morphism in $\mathop {\textbf {Bimod}}\nolimits$ whose image under $\Phi$ is the base-change morphism.

Appendix D. This appendix is devoted to the description of the reduced fiber of a flopping contraction $f \colon X\to Y$ with fibers of dimension bounded by one over a closed point of $Y$.

Appendix E. In this appendix we show that cohomology of an appropriate complex allow us to calculate morphisms in the derived category of an abelian category between bounded above complexes with bounded cohomology.

Notation

We denote by $k$ an algebraically closed field of characteristic zero. For a Noetherian $k$ scheme $X$, by $\mathop {{\rm Coh}}\nolimits (X)$, respectively ${\rm QCoh}(X)$, we denote the category of coherent, respectively quasi-coherent, sheaves on $X$.

For an abelian category ${\mathcal {A}}$, we denote by ${\mathcal {D}}^b({\mathcal {A}})$ and ${\mathcal {D}}({\mathcal {A}})$ the bounded and unbounded derived categories of ${\mathcal {A}}$. We write ${\mathcal {D}}^b(X) = {\mathcal {D}}^b(\mathop {{\rm Coh}}\nolimits (X))$, ${\mathcal {D}}_{{\rm qc}}(X) = {\mathcal {D}}({\rm QCoh}(X))$.

For an abelian category ${\mathcal {A}}$, by ${\rm Perf}({\mathcal {A}})$ we denote the full subcategory of ${\mathcal {D}}^b({\mathcal {A}})$ of objects that are quasi-isomorphic to finite complexes of projective objects in ${\mathcal {A}}$. For a scheme $X$, by ${\rm Perf}(X)$ we denote the category of perfect complexes on $X$, i.e. objects of ${\mathcal {D}}^b(X)$ locally quasi-isomorphic to finite complexes of locally free sheaves.

For a $k$-algebra $A$, we denote by ${\rm Mod}\unicode{x2013}A$ the abelian category of right $A$ modules.

For a $t$-structure $({\mathcal {T}}_{\leqslant 0}, {\mathcal {T}}_{\geqslant 0})$ on a triangulated category ${\mathcal {T}}$ with heart ${\mathcal {A}} = {\mathcal {T}}_{\leqslant 0 }\cap {\mathcal {T}}_{\geqslant 0}$ and an object $T \in {\mathcal {T}}$, we denote by ${\mathcal {H}}_{{\mathcal {A}}}^i(T)$ the $i$th cohomology of $T$ with respect to the $t$-structure $({\mathcal {T}}_{\leqslant 0}, {\mathcal {T}}_{\geqslant 0})$. The truncation functors are denoted by $\tau _{\geqslant i}^{{\mathcal {A}}}$ and $\tau ^{{\mathcal {A}}}_{\leqslant i}$. If ${\mathcal {T}} = {\mathcal {D}}_{{\rm qc}}(X)$ or ${\mathcal {D}}^b(X)$ with the standard $t$-structure with heart ${\rm QCoh}(X)$, respectively $\mathop {{\rm Coh}}\nolimits (X)$, we shorten the notation to ${\mathcal {H}}_X^i(T)$, $\tau _{\geqslant i}^X$ and $\tau _{\leqslant i}^X$ respectively.

Assumptions

Throughout the paper we assume $X$ and $Y$ to be normal varieties over $k$. We usually work under one of the following assumptions on a morphism $f\colon X\to Y$.

1. (a) We assume $f\colon X\to Y$ is a proper morphism with dimension of fibers bounded by one and such that $Rf_* {\mathcal {O}}_X \simeq {\mathcal {O}}_Y$.

2. (b) We assume $f\colon X\to Y$ is a projective birational morphism with dimension of fibers bounded by one between quasi-projective Gorenstein varieties of dimension $n\geqslant 3$. The exceptional locus of $f$ is of codimension greater than one in $X$. Variety $Y$ has canonical hypersurface singularities of multiplicity two.

3. (c) We assume the same as in assumption (p) and we further assume that variety $Y$ is affine and is embedded as a principal divisor into a smooth variety ${\mathcal {Y}}$ of dimension $n+1$.

4. (d) We assume the same as in assumption (p) with an extra assumption that $Y= \mathop {{\rm Spec}}\nolimits R$, where $R$ is a complete local $k$-algebra.

Let $f$ satisfy assumption (p). As $Y$ has canonical singularities, $Rf_* {\mathcal {O}}_X \simeq {\mathcal {O}}_Y$ (cf. [Elk81]), i.e. assumption (d) is satisfied when $Y$ has rational singularities and $f\colon X\to Y$ is a smooth birational resolution.

2. The null category $\mathscr {A}_f$

Let $f\colon X \to Y$ be a proper morphism of Noetherian schemes. In this section we introduce the null category of $f$:

(5)\begin{equation} \mathscr{A}_f = \{ E\in \mathop{{\rm Coh}}\nolimits(X)\,|\, Rf_*(E) = 0\}. \end{equation}

Under the assumption that the dimension of fibers of $f$ is bounded by one, $Y$ is affine and $Rf_*{\mathcal {O}}_X = {\mathcal {O}}_Y$, we construct a projective generator ${\mathcal {P}}$ for $\mathscr {A}_f$. We also study the behaviour of the null category under decomposition $f = h \circ g$ and under restriction to fibers.

2.1 The triangulated and abelian null category of a morphism of schemes

For a proper morphism of Noetherian schemes $f \colon X \to Y$, we consider the triangulated null category ${\mathcal {C}}_f$ defined as the kernel of functor $Rf_*$ restricted to ${\mathcal {D}}^b(X)$:

(6)\begin{equation} {\mathcal{C}}_f = \{ E^{\boldsymbol{\cdot}} \in {\mathcal{D}}^b(X)\,|\, Rf_*(E^{\boldsymbol{\cdot}}) = 0\}. \end{equation}

Similarly, we define ${\mathcal {C}}_f^- \subset {\mathcal {D}}^-(X)$ and ${{\mathcal {C}}_f}_{{\rm qc}} \subset {\mathcal {D}}_{{\rm qc}}(X)$. Denote by ${\iota _f}_* \colon {{\mathcal {C}}_f}_{{\rm qc}} \to {\mathcal {D}}_{{\rm qc}}(X)$ the inclusion functor. As $Lf^*\colon {\mathcal {D}}_{{\rm qc}}(Y) \to {\mathcal {D}}_{{\rm qc}}(X)$ is fully faithful with right adjoint $Rf_*$, ${\mathcal {D}}_{{\rm qc}}(X)$ admits a SOD ${\mathcal {D}}_{{\rm qc}}(X) = \langle {{\mathcal {C}}_f}_{{\rm qc}}, Lf^*{\mathcal {D}}_{{\rm qc}}(Y)\rangle$, [Bon89, Lemma 3.1]. In particular, ${\iota _f}_*$ admits a left adjoint $\iota _f^*\colon {\mathcal {D}}_{{\rm qc}}(X)\to {{\mathcal {C}}_f}_{{\rm qc}}$, which fits into a functorial exact triangle (see (B.1)):

(7)\begin{equation} Lf^* Rf_*\to \mathop{{\rm Id}}\nolimits_{{\mathcal{D}}_{{\rm qc}}(X)} \to \iota_{f*} \iota_f^* \to Lf^*Rf_*[1]. \end{equation}

For the null category of $f$ as in (5), we have $\mathscr {A}_f=\mathop {{\rm Coh}}\nolimits (X)\cap {\mathcal {C}}_f$.

Lemma 2.1 Let $f \colon X \to Y$ be a proper morphism with dimension of fibers bounded by one. Then category $\mathscr {A}_f$ is abelian. The embedding functor $\mathscr {A}_f \to \mathop {{\rm Coh}}\nolimits (X)$ is exact and fully faithful. Its image is closed under extensions.

Proof. Let $A, B$ be objects in $\mathscr {A}_f$ and let $\varphi \in \mathop {{\rm Hom}}\nolimits _X(A,B)$. Consider the kernel $K$, the cokernel $C$, and the image $I$ of $\varphi$. As $Rf_*A \simeq 0 \simeq Rf_* B$, the long exact sequences of higher derived functors for $f_*$ applied to short exact sequences on $X$

\[ 0 \to K \to A \to I\to 0,\quad 0 \to I \to B \to C \to 0, \]

together with vanishing of $R^if_*$, for $i>1$, imply that first $I$, hence $K$ and $C$ lie in $\mathscr {A}_f$. That $\mathscr {A}_f$ is closed under extensions is obvious.

It follows that $\mathop {{\rm Ext}}\nolimits ^1_X(A,B)=\mathop {{\rm Ext}}\nolimits ^1_{\mathscr {A}_f}(A,B)$, for $A, B \in \mathscr {A}_f$.

Lemma 2.2 [Bri02, Lemma 3.1]

Let $X$ and $Y$ be Noetherian schemes and $f \colon X \to Y$ a proper morphism with fibers of dimension bounded by one. Then $E\in {\mathcal {C}}_f$ if and only if ${\mathcal {H}}^i_X(E)\in \mathscr {A}_f$, for all $i\in \mathbb {Z}$. In particular, $\mathscr {A}_f \subset {\mathcal {C}}_f$ is the heart of a bounded $t$-structure.

Remark 2.3 The restriction of the standard $t$-structure on ${\mathcal {D}}_{{\rm qc}}(X)$ to

\[ {{\mathcal{C}}_f}_{{\rm qc}} := \{ E^{\boldsymbol{\cdot}}\in {\mathcal{D}}_{{\rm qc}}(X)\,|\, Rf_*(E^{\boldsymbol{\cdot}}) =0 \} \]

was considered by Bridgeland [Bri02] for a birational morphism of projective varieties satisfying assumption (d) which is easily generalised to a projective morphism $f\colon X\to Y$ satisfying assumption (d). In this case, category ${\mathcal {D}}_{{\rm qc}}(X)$ admits a recollement [BBD82] or, equivalently, an admissible subcategory, ${{\mathcal {C}}_f}_{{\rm qc}}$ (see [Bon89]). The Verdier quotient ${\mathcal {D}}_{{\rm qc}}(X)/ {{\mathcal {C}}_f}_{{\rm qc}}$ is identified with ${\mathcal {D}}_{{\rm qc}}(Y)$. The $t$-structure on ${{\mathcal {C}}_f}_{{\rm qc}}$ with heart $\mathscr {A}_f[-p]$ can be glued with the standard $t$-structure on ${\mathcal {D}}_{{\rm qc}}(Y)$ (see [BBD82]). As functor $f^!$ is needed for the gluing of $t$-structures, one has to consider unbounded derived categories of quasi-coherent sheaves on $X$ and $Y$. The heart of the resulting $t$-structure is the category ${}^p{\rm Per}_{{\rm qc}}(X/Y)$ of perverse sheaves. If $p=-1$ or $p=0$, the $t$-structures with hearts ${}^p{\rm Per}_{{\rm qc}}(X/Y)$ can also be obtained by the tilting in torsion pairs $({\mathcal {T}}_{-1}, {\mathcal {F}}_{-1})$, $({\mathcal {T}}_{0}, {\mathcal {F}}_{0})$ in ${\rm QCoh(X)}$ (see [VdB04]), where

(8)\begin{align} {\mathcal{T}}_0&= \{ T \in \mathop{{\rm Coh}}\nolimits(X) \, |\, R^1 f_*(T)=0 \}, \end{align}

(9)\begin{align} {\mathcal{F}}_0& = \{ E \in \mathop{{\rm Coh}}\nolimits(X)\,|\, f_*(E) = 0,\, \mathop{{\rm Hom}}\nolimits(\mathscr{A}_f, E) = 0 \}, \end{align}

(10)\begin{align} {\mathcal{T}}_{-1}& = \{T \in \mathop{{\rm Coh}}\nolimits(X)\,|\, R^1f_*(T) = 0,\, \mathop{{\rm Hom}}\nolimits(T, \mathscr{A}_f) = 0\}, \end{align}

(11)\begin{align} {\mathcal{F}}_{-1}& = \{ E \in \mathop{{\rm Coh}}\nolimits(X)\,|\, f_*(E) =0 \}. \end{align}

By restricting the torsion pairs to $\mathop {{\rm Coh}}\nolimits (X)$ one can define abelian categories ${}^p{\rm Per}(X/Y)$ as subcategories of ${\mathcal {D}}^{b}(X)$. Thus, for $p=-1,0$,

\[ \mathop{{}^{p}{\rm Per}}\nolimits(X/Y) := \{ E\in {\mathcal{D}}^b(X)\,|\, {\mathcal{H}}^0(E) \in {\mathcal{T}}_p,\, {\mathcal{H}}^{-1}(E) \in {\mathcal{F}}_p \text{ and }{\mathcal{H}}^i(E) =0,\ \text{for }\, i\neq -1,0\}.\]

2.2 A projective generator for $\mathscr {A}_f$

Let $f\colon X\to Y$ satisfy assumption (d) and assume $Y$ is affine. By [VdB04, Proposition 3.2.5], there exists a vector bundle ${\mathcal {M}}$ on $X$ which is a projective generator for $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$. Consider

(12)\begin{equation} {\mathcal{P}} := {\mathcal{H}}^{-1}_X \iota_f^* {\mathcal{M}}. \end{equation}

It is an object in $\mathscr {A}_f$, which, by abuse of notation, we identify with its image under $\iota _{f*}$.

Proposition 2.4 Sheaf ${\mathcal {P}}$ is a projective generator for the category $\mathscr {A}_f$.

Proof. As the $t$-structure on ${{\mathcal {C}}_f}_{{\rm qc}}$ is defined as the restriction of the standard $t$-structure on ${\mathcal {D}}_{{\rm qc}}(X)$, the functor $\iota _{f*} \colon {{\mathcal {C}}_f}_{{\rm qc}} \to {\mathcal {D}}_{\rm qc}(X)$ is exact, in particular, it commutes with the truncation functors $\tau _{\geqslant i}$. This implies isomorphism of functors

\[ \tau^X_{\geqslant -1} \iota_{f*} \simeq \iota_{f*} \tau^{\mathscr{A}_f}_{\geqslant -1}. \]

It follows that $\tau ^X_{\geqslant -1} \iota _{f*} \iota _f^* {\mathcal {M}}$ is isomorphic to $\iota _{f*}\circ {}^{-1}\iota _f^* {\mathcal {M}}$, where

\[ {}^{-1} \iota_f^* := {\mathcal{H}}^{-1}_{\mathscr{A}_f} \circ \iota_f^* \colon \mathop{{}^{-1}{\rm Per}}\nolimits(X/Y) \to \mathscr{A}_f[1] \]

is the left adjoint functor to the inclusion $\mathscr {A}_f[1] \to \mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$. By Lemma 2.5, ${\mathcal {P}}\in \mathscr {A}_f$ is a projective generator.

Lemma 2.5 Let ${\mathcal {D}}_0$ and ${\mathcal {D}}$ be triangulated categories with $t$-structures with hearts ${\mathcal {A}}_0\subset {\mathcal {D}}_0$ and ${\mathcal {A}}\subset {\mathcal {D}}$, and $i_*\colon {\mathcal {D}}_0 \to {\mathcal {D}}$ a $t$-exact functor with left adjoint $i^*$. If ${\mathcal {M}} \in {\mathcal {A}}$ is projective, then ${\mathcal {P}}:={\mathcal {H}}^0(i^* {\mathcal {M}}) \in {\mathcal {A}}_0$ is projective. Moreover, if ${\mathcal {M}} \in {\mathcal {A}}$ is a projective generator and $i_*$ is fully faithful, then ${\mathcal {P}} \in {\mathcal {A}}_0$ is a projective generator.

Proof. Let $E$ be an object in ${\mathcal {A}}_0$. Then $i_*E \in {\mathcal {A}}$, hence $0=\mathop {{\rm Ext}}\nolimits ^1_{{\mathcal {D}}}({\mathcal {M}}, i_*E) \simeq \mathop {{\rm Ext}}\nolimits ^1_{{\mathcal {D}}_0}(i^*{\mathcal {M}}, E)$. As functor $i^*$ is left adjoint to a $t$-exact functor, it is right $t$-exact, i.e. $i^* {\mathcal {M}} \in {\mathcal {D}}_0^{\leqslant 0}$. Thus, we have an exact triangle

(13)\begin{equation} \tau_{\leqslant -1} i^* {\mathcal{M}} \to i^* {\mathcal{M}} \to {\mathcal{P}} \to \tau_{\leqslant -1}i^* {\mathcal{M}}[1]. \end{equation}

As, for degree reasons, $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}_0}(\tau _{\leqslant -1}i^*M, E) =0$, by applying $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}_0}(-,E)$ to (13), we get that $\mathop {{\rm Ext}}\nolimits ^1_{{\mathcal {D}}_0}({\mathcal {P}}, E)=0$, i.e. ${\mathcal {P}}\in {\mathcal {A}}_0$ is projective.

If ${\mathcal {M}}\in {\mathcal {A}}$ is a projective generator and $i_*$ is fully faithful, then, for any non-zero $E\in {\mathcal {A}}_0$, object $i_*E$ is non-zero, hence $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}}({\mathcal {M}}, i_*E) \simeq \mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}_0}(i^*{\mathcal {M}}, E)\neq 0$. It follows from the long exact sequence obtained by applying $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}_0}(-,E)$ to (13) that $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}_0}({\mathcal {P}}, E) \neq 0$, i.e. ${\mathcal {P}}$ is a projective generator for ${\mathcal {A}}_0$.

Remark 2.6 If morphism $f$ satisfies assumption (d) and $Y$ is affine, the category $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$ has a projective generator ${\mathcal {N}} = {\mathcal {M}}^{\vee }$ (see [VdB04, Proposition 3.2.5]). An argument analogous to that in the proof of Proposition 2.4 shows that ${\mathcal {P}}_{{\mathcal {N}}} := {\mathcal {H}}_X^0 \iota _f^* {\mathcal {N}}$ is a projective generator for $\mathscr {A}_f$.

For a sheaf $F$ with $R^1f_*F=0$, the cohomology sequence of triangle (7) applied to $F$ gives an exact sequence

(14)\begin{equation} 0 \to {\mathcal{H}}_X^{-1}(\iota_{f_*}\iota_f^* F) \to f^*f_* F\to F\to {\mathcal{H}}_X^0(\iota_{f*}\iota_f^* F) \to 0. \end{equation}

As by [VdB04, Lemma 3.1.3] morphism $f^*f_* {\mathcal {M}} \to {\mathcal {M}}$ is surjective, sequence

(15)\begin{equation} 0 \to {\mathcal{P}} \to f^*f_* {\mathcal{M}} \to {\mathcal{M}} \to 0 \end{equation}

is exact in $\mathop {{\rm Coh}}\nolimits (X)$.

Let $f\colon X\to Y$ satisfy assumption (d) and $Y = \mathop {{\rm Spec}}\nolimits R$ be a spectrum of a complete Noetherian local ring $R$. The reduced fiber $C_{{\rm red}} = \bigcup _{i=1}^n C_i$ of $f$ over the unique closed point $y\in Y$ is a tree of rational curves (see a more precise statement in Theorem D.1).

The Picard group of $X$ is isomorphic to $\mathbb {Z}^n$, where the isomorphism is given by the degrees of the restriction to irreducible components of $C_{{\rm red}}$: ${\mathcal {L}} \mapsto \deg ({\mathcal {L}}|_{C_i})_{i=1,\ldots,n}$.

Remark 2.7 (Cf. [VdB04, Lemma 3.4.4])

Let $x_i \in C_i\subset X$ be a closed point such that $x_i \notin C_k$, for any $k \neq i$, and $j_i\colon {X}_i \to \mathcal {X}_i$ a closed embedding of a neighbourhood ${X}_i$ of $x_i$ in $X$ into a smooth variety $\mathcal {X}_i$. There exists an effective Cartier divisor $\mathcal {D}_i \subset \mathcal {X}_i$ such that scheme-theoretically $\mathcal {D}_i \cap j_{i*} C_i = \{j_{i*} x_i\}$. By pulling back $\mathcal {D}_i$ to ${X}_i$, we obtain an effective divisor $D_i \subset X$ such that scheme-theoretically $D_i.C_i = \{x_i\}$ and $D_i.C_k =0$, for $k\neq i$. We denote by $\iota _{D_i} \colon D_i \to X$ the embedding of $D_i$ into $X$.

Denote by ${\mathcal {L}}_i$ the line bundle on $X$ defined by

(16)\begin{equation} {\mathcal{L}}_i := {\mathcal{O}}_X(D_i). \end{equation}

For every $i$, Van den Bergh defined a vector bundle ${\mathcal {M}}_i$ via the exact sequence

(17)\begin{equation} 0 \to {\mathcal{O}}_X^{r_i-1} \to {\mathcal{M}}_i \to {\mathcal{L}}_i \to 0 \end{equation}

corresponding to a choice of $r_i-1$ generators of $\mathop {{\rm Ext}}\nolimits ^1_X({\mathcal {L}}_i, {\mathcal {O}}_X)$ as an $R$-module. Denote ${\mathcal {M}}_0:={\mathcal {O}}_X$. Then

\[ {\mathcal{M}} = \bigoplus_{i=0}^n {\mathcal{M}}_i \]

is a projective generator for ${}^{-1}{\rm Per}(X/Y)$ (see [VdB04, Proposition 3.5.4]). The dual vector bundle

(18)\begin{equation} {\mathcal{N}} = {\mathcal{M}}^{\vee} \end{equation}

is a projective generator for $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$ (see [VdB04, Proposition 3.2.5]).

We put

(19)\begin{equation} {\mathcal{P}}_i = {\mathcal{H}}_X^{-1} \iota_f^* {\mathcal{M}}_i. \end{equation}

By Proposition 2.4, sheaf ${\mathcal {P}} = \bigoplus _{i=1}^n {\mathcal {P}}_i$ is a projective generator for $\mathscr {A}_f$.

Example Let $X$ be a smooth threefold, $f\colon X\to Y$ a flopping contraction. Assume $Y = \mathop {{\rm Spec}}\nolimits R$ has rational singularities and $C_{{\rm red}}\simeq \mathbb {P}^1\subset X$. Then the normal bundle $N_{X/C_{{\rm red}}}$ is isomorphic either to ${\mathcal {O}}(-1) \oplus {\mathcal {O}}(-1)$, ${\mathcal {O}} \oplus {\mathcal {O}} (-2)$ or ${\mathcal {O}} (1) \oplus {\mathcal {O}} (-3)$. Let $D\subset X$ be a divisor such that $D.C_{{\rm red}} = 1$. For the first two cases, ${\mathcal {M}} \simeq {\mathcal {O}}_X \oplus {\mathcal {O}}_X(D)$, and it is of higher rank for the third case. If $N_{X/C_{{\rm red}}} \simeq {\mathcal {O}} (-1) \oplus {\mathcal {O}} (-1)$, the projective generator ${\mathcal {P}}$ is ${\mathcal {O}} (-1)$. When $N_{X/C_{{\rm red}}} \simeq {\mathcal {O}} \oplus {\mathcal {O}} (-2)$, ${\mathcal {P}}$ is an $n$-iterated extension of ${\mathcal {O}} (-1)$ by ${\mathcal {O}} (-1)$, where $n$ is the width of $C_{{\rm red}}$, see [Rei83] (cf. [Tod07]).

2.3 Basic properties of $\mathscr {A}_f$

Let $f \colon X \to Y$ satisfy assumption (d) and let $E$ be a coherent sheaf on $Y$. By the derived projection formula, we have $Rf_*Lf^*(E) \simeq E$. As $f$ has fibers of dimension bounded by one, Leray spectral sequence $R^pf_* L^qf^*(E) \Rightarrow {\mathcal {H}}_X^{p-q} E$ degenerates. Hence, an exact sequence:

(20)\begin{equation} 0 \to R^1f_*L^1f^*E \to E \to f_*f^* E \to 0. \end{equation}

Lemma 2.8 Let $f \colon X \to Y$ and $E$ be as above. Then $R^1f_*f^*E = 0$ and $Rf_*L^if^*E = 0$, for any $i >1$. Further suppose that $E$ has no torsion supported at the image $f({\rm Ex}\, f)$ of the exceptional locus of $f$. Then $f_*f^*(E) \simeq E$ and sheaf $L^1f^*E$ is in $\mathscr {A}_f$.

Proof. The derived projection formula $Rf_*Lf^* E \simeq E$ and Leray spectral sequence $R^pf_* L^qf^*(E) \Rightarrow {\mathcal {H}}_X^{p-q} E$ imply $R^1f_*f^* E \simeq 0$, $f_*L^1f^* E \simeq 0$, $Rf_* L^if^* E \simeq 0$, for any $i>1$. As $R^1f_*L^1f^* E$ is supported on $f({\rm Ex}\, f)$, the assumption that $E$ has no torsion supported on $f(\mathop {{\rm Ex}}\nolimits f)$ and sequence (20) imply that $R^1f_*L^1f^*E$ is zero and $E \simeq f_*f^*E$.

Lemma 2.9 Let $f\colon X \to Y$ and $E$ be as above. If $E=f_* E'$, for some $E'\in \mathop {{\rm Coh}}\nolimits (X)$, then $f_*f^*(E) \simeq E$ and sheaves $L^if^*E$ are in $\mathscr {A}_f$, for $i>0$.

Proof. Sequence (20) implies that morphism $\alpha \colon f_* E' \to f_*f^*f_* E'$ is surjective. Morphism $\alpha$ is the inclusion of a direct summand, because its composition with the canonical map $f_*f^*f_* E'\to f_* E'$ is the identity. Hence, $\alpha$ is an isomorphism. Exact sequence (20) implies that $R^1f_*L^1f^*f_* E' = 0$. The rest follows from Lemma 2.8.

Proposition 2.10 Let $f\colon X\to Y$ be a proper morphism with dimension of fibers bounded by one. Consider a decomposition for $f$:

Then, for $E \in \mathop {{\rm Coh}}\nolimits (X)$ with $R^1f_* E =0$, we have $R^1g_* E = 0$. Functor $g_*$ restricts to an exact functor $g_*\colon \mathscr {A}_f \to \mathscr {A}_h$.

Proof. Morphism $g$ is proper by the valuative criterion. Replace $Z$ by the (closed) image of $g$, if necessary. Then $h$ becomes a proper morphism, cf. [Gro61, Corollaire 5.4.3]. As the dimension of fibers for $f$ is bounded by one, so is the dimension of fibers for $h$.

As $Rf_*(E)$ is a sheaf on $Y$, spectral sequence $R^qh_* R^sg_* E \Rightarrow R^{q+s}f_* E$ implies that $Rh_*R^1g_*(E) = 0$. Sheaf $R^1g_*(E)$ is supported in the locus of points $z \in Z$ such that the fiber of morphism $g$ over $z$ is one-dimensional. As the null category of a finite morphism is zero, morphism $h\colon Z\to Y$ restricted to the support of $R^1g_*(E)$ must have fibers of dimension one. Let $y \in Y$ be a point in $h({\rm Supp}\, R^1g_*(E))$. Then the fiber of $f$ over $y$ is two-dimensional, which contradicts the assumptions. Thus, $R^1g_*(E)$ is zero.

For any $E \in \mathscr {A}_f$, the spectral sequence $R^ph_*R^qg_* E \Rightarrow R^{p+q}f_* E = 0$ degenerates. It follows that $g_* E \in \mathscr {A}_h$. As $R^1g_*(E) = 0$, functor $g_* \colon \mathscr {A}_f \to \mathscr {A}_h$ is exact.

Proposition 2.11 Let $f\colon X\to Y$ be a proper morphism with dimension of fibers bounded by one and $g \colon Z \to Y$ a morphism of schemes over field $k$. Assume a coherent sheaf $E$ on $X$ satisfy $R^lf_* E = 0$, for $l\geqslant l_0$, for some $l_0 \in \{0,1\}$. Then $R^l\pi _{Z*} \pi _X^* E = 0$, for $l\geqslant l_0$, where $\pi _X \colon W \to X$ and $\pi _Z \colon W \to Z$ are the projections for $W = X\times _Y Z$:

(21)

Proof. The construction of $X\times _Y Z$ and the statement are local on $Y$, hence we may assume that $Y=\mathop {{\rm Spec}}\nolimits (R)$ is affine. Moreover, the statement is local on $Z$, so we may further reduce to the case when $Z = \mathop {{\rm Spec}}\nolimits (A)$ is affine. Hence, $g\colon Z\to Y$ is an affine morphism, which implies that $\pi _X$ is affine.

The commutativity of (21) implies $R(f\pi _X)_*\pi _X^* E \simeq R(g\pi _Z)_*\pi _X^* E$. Morphisms $g$ and $\pi _X$ are affine, hence we have an isomorphism $R^lf_* (\pi _{X*}\pi _X^* E) \simeq g_*R^l\pi _{Z*}(\pi _X^* E)$, for any $l\geqslant 0$. Morphism $g$ is affine, thus $g_*R^l\pi _{Z*}(\pi _X^*E)$ is zero if and only if $R^l\pi _{Z*}(\pi _X^* E)$ is also zero. Hence, we need to show that $R^lf_* (\pi _{X*}\pi _X^* E) = 0$, for $l\geqslant l_0$.

Morphism $\pi _X$ is affine, hence $\pi _{X*}\pi _X^* E \simeq E \otimes \pi _{X*}({\mathcal {O}}_{X\times _Y Z})$. The base-change morphism (cf. (47)) $f^*g_* {\mathcal {O}}_{Z}\to \pi _{X*} {\mathcal {O}}_{X\times _Y Z}$ is an isomorphism. Indeed, this can be checked locally on $X$: if $X = \mathop {{\rm Spec}}\nolimits (B)$, then both sheaves $\pi _{X*} {\mathcal {O}}_{X\times _Y Z}$ and $f^*g_* {\mathcal {O}}_Z$ correspond to $B$-module $B\otimes _R A$. Thus,

\[ \pi_{X*}\pi_X^* E \simeq E \otimes f^*g_* {\mathcal{O}}_Z \simeq {\mathcal{H}}^0_X(E \otimes^L Lf^* g_* {\mathcal{O}}_Z). \]

Derived projection formula $Rf_*(E \otimes ^L Lf^*g_* {\mathcal {O}}_Z) \simeq Rf_*(E) \otimes ^L g_*{\mathcal {O}}_Z$ implies that $Rf_* (E \otimes ^L Lf^* g_* {\mathcal {O}}_Z) \in {\mathcal {D}}(Y)^{< l_0}$. Morphism $f$ is proper and with dimension of fibers bounded by one, hence $R^pf_*(F)=0$, for $p>1$ and for any sheaf $F$. It follows that spectral sequence

\[ R^pf_*{\mathcal{H}}_X^q(E \otimes^L Lf^*g_* {\mathcal{O}}_Z) \Rightarrow R^{p+q}f_*(E \otimes^L Lf^*g_* {\mathcal{O}}_Z) \]

degenerates. Therefore, $R^lf_*(\pi _{X*}\pi _X^* E) \simeq R^l f_* {\mathcal {H}}_X^0(E \otimes Lf^*g_* {\mathcal {O}}_Z) = 0$, for $l\geqslant l_0$.

Corollary 2.12 Let $f\colon X\to Y$ and $g \colon Z \to Y$ be as in Proposition 2.11. For $E\in \mathscr {A}_f$, its pull-back $\pi _X^* E$ is an object in $\mathscr {A}_{\pi _Z}$.

3. $L^1f^*$ vanishing and 2-periodicity

Let $f\colon X\to Y$ satisfy assumption (d). For $p=-1,0$, we denote by $\mathscr {P}_p$ the category of locally projective objects in $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$. An object ${\mathcal {M}}$ belongs to $\mathscr {P}_p$ if there exists an affine open covering $Y = \bigcup Y_i$, inducing $X = \bigcup X_i$ with $X_i = f^{-1}(Y_i)$, such that ${\mathcal {M}}|_{X_i}$ is projective in $\mathop {{}^{p}{\rm Per}}\nolimits (X_i/Y_i)$. By [VdB04, Proposition 3.2.6] objects in $\mathscr {P}_{-1}$ and $\mathscr {P}_0$ are locally free sheaves on $X$.

In this section we discuss a spherical couple associated to a Cartier divisor. Under the assumption that $Y$ has hypersurface singularities we prove that the sheaf $L^1f^*f_* {\mathcal {M}}$ is zero, for any object ${\mathcal {M}} \in \mathscr {P}_{-1}$. Though technical, this result is crucial for the various description of the flop and flop–flop functors presented in the subsequent sections.

3.1 Cartier divisors and spherical couples

Let $X$ be a quasi-compact, quasi-separated scheme. By an effective Cartier divisor $i\colon D\to X$ we mean a subscheme whose ideal sheaf ${\mathcal {I}}_D$ is invertible. Here we discuss a 2-categorical adjunction and a spherical couple in the bicategory $\mathop {\textbf {FM}}\nolimits$ (see Appendix C) related to such a divisor.

Denote by $\Gamma \subset D\times X$, $\Gamma ^t \subset X\times D$ the graphs of $i$. Sheaf ${\mathcal {O}}_\Gamma \in {\mathcal {D}}_{{\rm qc}}(D\times X)$ is an FM kernel for $Ri_*\colon {\mathcal {D}}_{{\rm qc}}(D) \to {\mathcal {D}}_{{\rm qc}}(X)$, and ${\mathcal {O}}_{\Gamma ^t}\in {\mathcal {D}}_{{\rm qc}}(X\times D)$ is an FM kernel for $Li^*\colon {\mathcal {D}}_{{\rm qc}}(X) \to {\mathcal {D}}_{{\rm qc}}(D)$.

For a scheme $Y$, we denote by $\Delta ^Y\subset Y\times Y$ the diagonal and, for closed $Z\subset Y$, by $\Delta ^{Z,Y}$ the image of $Z$ under the diagonal morphism ${\rm diag}^Y\colon Y \to Y \times Y$. For $F\in {\mathcal {D}}_{{\rm qc}}(D\times Y)$, we have

(22)\begin{equation} F \ast {\mathcal{O}}_{\Gamma^t} = (i\times \mathop{{\rm Id}}\nolimits_Y)_*F\in {\mathcal{D}}_{{\rm qc}}(X\times Y). \end{equation}

In particular,

\[ {\mathcal{O}}_{\Gamma} \ast {\mathcal{O}}_{\Gamma^t} = (i\times \mathop{{\rm Id}}\nolimits_X)_*{\mathcal{O}}_{\Gamma} \simeq {\mathcal{O}}_{\Delta^{D,X}}. \]

Embedding $\Delta ^{D,X}\subset \Delta ^X$ gives

(23)\begin{equation} \eta \colon {\mathcal{O}}_{\Delta^X} \to {\mathcal{O}}_{\Delta^{D,X}} \simeq {\mathcal{O}}_{\Gamma^t} \ast {\mathcal{O}}_{\Gamma}. \end{equation}

For any scheme $Y$, and objects $F\in {\mathcal {D}}_{{\rm qc}}(X\times Y)$, $G\in {\mathcal {D}}_{{\rm qc}}(Y \times X)$, we have

\[ F \ast {\mathcal{O}}_{\Gamma} \ast {\mathcal{O}}_{\Gamma^t} \simeq (i\times\mathop{{\rm Id}}\nolimits_Y)_*(i\times \mathop{{\rm Id}}\nolimits_Y)^*F,\quad {\mathcal{O}}_{\Gamma} \ast {\mathcal{O}}_{\Gamma^t} \ast G \simeq (\mathop{{\rm Id}}\nolimits_Y \times i)_*(\mathop{{\rm Id}}\nolimits_Y\times i)^*G. \]

One checks locally that for $F\in {\rm QCoh}(X\times Y)$ and $G\in {\rm QCoh}(Y\times X)$ morphisms

\begin{align*} &F\simeq F\ast {\mathcal{O}}_{\Delta^X} \xrightarrow{F \ast \eta } F \ast {\mathcal{O}}_{\Gamma} \ast {\mathcal{O}}_{\Gamma^t} \simeq (i\times \mathop{{\rm Id}}\nolimits_Y)_*(i\times \mathop{{\rm Id}}\nolimits_Y)^*F \simeq F\otimes {\mathcal{O}}_{D\times Y},\\ &G \simeq {\mathcal{O}}_{\Delta^X} \ast G \xrightarrow{\eta\ast G }{\mathcal{O}}_{\Gamma} \ast {\mathcal{O}}_{\Gamma^t} \ast G \simeq (\mathop{{\rm Id}}\nolimits_Y\times i)_*(\mathop{{\rm Id}}\nolimits_Y\times i)^*G \simeq G\otimes {\mathcal{O}}_{Y\times D} \end{align*}

are induced by restriction morphisms ${\mathcal {O}}_{X\times Y} \to {\mathcal {O}}_{D \times Y}$ and ${\mathcal {O}}_{Y\times X} \to {\mathcal {O}}_{Y \times D}$, respectively.

For any scheme $Y$ and $F\in {\mathcal {D}}_{{\rm qc}}(X\times Y)$, we have

(24)\begin{equation} F \ast {\mathcal{O}}_{\Gamma} \simeq (i\times\mathop{{\rm Id}}\nolimits_Y)^*F\in {\mathcal{D}}_{{\rm qc}}(D\times Y). \end{equation}

Hence,

\[ {\mathcal{O}}_{\Gamma^t} \ast {\mathcal{O}}_{\Gamma} \simeq (i\times \mathop{{\rm Id}}\nolimits_D)^*{\mathcal{O}}_{\Gamma^t} \simeq (i\times \mathop{{\rm Id}}\nolimits_D)^*(i\times \mathop{{\rm Id}}\nolimits_D)_*{\mathcal{O}}_{\Delta^D} \in {\mathcal{D}}_{{\rm qc}}(D\times D). \]

Subscheme $D\times D\subset X\times D$ is a Cartier divisor with ideal sheaf ${\mathcal {I}}_{D\times D}$. Therefore, object $(i\times \mathop {{\rm Id}}\nolimits _D)^*(i\times \mathop {{\rm Id}}\nolimits _D)_*{\mathcal {O}}_{\Delta ^D}$ has two non-zero cohomology sheaves, ${\mathcal {O}}_{\Delta ^D}$ in degree $0$ and ${\mathcal {O}}_{\Delta ^D}\otimes {\mathcal {I}}_{D\times D}|_{D\times D}$ in degree $-1$. Truncation to the zeroth cohomology gives morphism

(25)\begin{equation} \varepsilon \colon {\mathcal{O}}_{\Gamma^t} \ast {\mathcal{O}}_{\Gamma} \simeq (i\times\mathop{{\rm Id}}\nolimits_D)^*(i\times\mathop{{\rm Id}}\nolimits_D)_*{\mathcal{O}}_{\Delta^D} \to {\mathcal{O}}_{\Delta^D} \end{equation}

Formulas (22) and (24) imply that, for any scheme $Y$ and objects $F\in {\mathcal {D}}_{{\rm qc}}(D\times Y)$ and $G\in {\mathcal {D}}_{{\rm qc}}(Y\times D)$, we have

\[ F \ast {\mathcal{O}}_{\Gamma^t} \ast {\mathcal{O}}_{\Gamma} \simeq (i\times \mathop{{\rm Id}}\nolimits_Y)^*(i\times\mathop{{\rm Id}}\nolimits_Y)_*F, \quad {\mathcal{O}}_{\Gamma^t} \ast {\mathcal{O}}_{\Gamma} \ast G \simeq (\mathop{{\rm Id}}\nolimits_Y\times i)^*(\mathop{{\rm Id}}\nolimits_Y\times i)_*G. \]

One checks locally along $D$ that for $F\in {\rm QCoh}(D\times Y)$ and $G\in {\rm QCoh}(Y\times D)$ morphisms

\begin{align*} (i\times\mathop{{\rm Id}}\nolimits_Y)^*(i\times\mathop{{\rm Id}}\nolimits_Y)_*F &\simeq F \ast {\mathcal{O}}_{\Gamma^t} \ast {\mathcal{O}}_{\Gamma} \xrightarrow{F \ast \varepsilon } F \ast {\mathcal{O}}_{\Delta^D} \simeq F,\\ (\mathop{{\rm Id}}\nolimits_Y\times i)^*(\mathop{{\rm Id}}\nolimits_Y\times i)_*G &\simeq {\mathcal{O}}_{\Gamma^t} \ast {\mathcal{O}}_{\Gamma} \ast G \xrightarrow{\varepsilon\ast G}{\mathcal{O}}_{\Delta^D}\ast G \simeq G \end{align*}

are the truncations to the zeroth cohomology.

An early version of formula (26) in the following theorem was first proven in [BO95, Lemma 3.3] under the assumption that both varieties are smooth. The result was also stated without proof in [Ann07, Add16].

Theorem 3.1 Let $i\colon D \to X$ be the embedding of an effective Cartier divisor. Then $({\mathcal {O}}_{\Gamma ^t}, {\mathcal {O}}_{\Gamma }, \eta, \varepsilon )$ is a spherical couple in the bicategory $\mathop {\textbf {FM}}\nolimits$. Object ${\rm diag}^D_*{\mathcal {I}}_D|_D[2]$ is the spherical twist and ${\rm diag}^X_*{\mathcal {I}}_D$ the spherical cotwist. The associated functorial exact triangles read

(26)\begin{align} &\mathop{{\rm Id}}\nolimits_{{\mathcal{D}}_{{\rm qc}}(X)} \otimes {\mathcal{I}}_D \to \mathop{{\rm Id}}\nolimits_{{\mathcal{D}}_{{\rm qc}}(X)} \xrightarrow{\eta} Ri_*Li^* \to \mathop{{\rm Id}}\nolimits_{{\mathcal{D}}_{{\rm qc}}(X)} \otimes {\mathcal{I}}_D[1], \end{align}

(27)\begin{align} &{\rm Id}_{{\mathcal{D}}_{{\rm qc}}(D)}\otimes {\mathcal{I}}_D|_D[1] \to Li^*Ri_* \xrightarrow{\varepsilon} {\rm Id}_{{\mathcal{D}}_{{\rm qc}}(D)} \to {\rm Id}_{{\mathcal{D}}_{{\rm qc}}(D)}\otimes {\mathcal{I}}_D|_{D}[2]. \end{align}

Proof. First, we check that $({\mathcal {O}}_{\Gamma ^t},{\mathcal {O}}_{\Gamma }, \eta, \varepsilon )$ is a 2-categorical adjunction, i.e. that (C.1) are equal to the identity morphisms.

As ${\mathcal {O}}_{\Gamma } \simeq (\mathop {{\rm Id}}\nolimits _D\times i)_*{\mathcal {O}}_{\Delta ^D}$, the composite ${\mathcal {O}}_{\Gamma }\xrightarrow {\eta \ast {\mathcal {O}}_{\Gamma } } {\mathcal {O}}_{\Gamma } \ast {\mathcal {O}}_{\Gamma ^t} \ast {\mathcal {O}}_{\Gamma } \xrightarrow { {\mathcal {O}}_{\Gamma }\ast \varepsilon } {\mathcal {O}}_{\Gamma }$ reads

\[ (\mathop{{\rm Id}}\nolimits_D\times i)_*{\mathcal{O}}_{\Delta^D}\xrightarrow{\eta \ast {\mathcal{O}}_{\Gamma} } (\mathop{{\rm Id}}\nolimits_D\times i)_*(\mathop{{\rm Id}}\nolimits_D\times i)^*(\mathop{{\rm Id}}\nolimits_D\times i)_*{\mathcal{O}}_{\Delta^D} \xrightarrow{{\mathcal{O}}_{\Gamma}\ast \varepsilon } (\mathop{{\rm Id}}\nolimits_D\times i)_*{\mathcal{O}}_{\Delta^D}. \]

Object $(\mathop {{\rm Id}}\nolimits _D\times i)_*{\mathcal {O}}_{\Delta ^D}$ is supported on $D\times D$, hence the first morphism, restriction to $D\times D$, is the identity on the zeroth cohomology. The second morphism is the truncation at zeroth cohomology, hence $({\mathcal {O}}_\Gamma \ast \varepsilon )\circ (\eta \ast {\mathcal {O}}_{\Gamma } ) = \mathop {{\rm Id}}\nolimits _{{\mathcal {O}}_{\Gamma }}$.

As ${\mathcal {O}}_{\Gamma ^t} \simeq (i\times \mathop {{\rm Id}}\nolimits _D)_*{\mathcal {O}}_{\Delta ^D}$, the composite ${\mathcal {O}}_{\Gamma ^t}\xrightarrow { {\mathcal {O}}_{\Gamma ^t}\ast \eta } {\mathcal {O}}_{\Gamma ^t} \ast {\mathcal {O}}_{\Gamma } \ast {\mathcal {O}}_{\Gamma ^t} \xrightarrow {\varepsilon \ast {\mathcal {O}}_{\Gamma ^t} } {\mathcal {O}}_{\Gamma ^t}$ reads

\[ (i \times \mathop{{\rm Id}}\nolimits_D)_*{\mathcal{O}}_{\Delta^D}\xrightarrow{{\mathcal{O}}_{\Gamma^t}\ast \eta } (i \times \mathop{{\rm Id}}\nolimits_D)_*(i \times \mathop{{\rm Id}}\nolimits_D)^*(i \times \mathop{{\rm Id}}\nolimits_D)_*{\mathcal{O}}_{\Delta^D} \xrightarrow{\varepsilon \ast {\mathcal{O}}_{\Gamma^t} } (i\times \mathop{{\rm Id}}\nolimits_D)_*{\mathcal{O}}_{\Delta^D}. \]

The arguments as above show that $(\varepsilon \ast {\mathcal {O}}_{\Gamma ^t} ) \circ ( {\mathcal {O}}_{\Gamma ^t}\ast \eta )$ is the identity on ${\mathcal {O}}_{\Gamma ^t}$.

Now we calculate the twist and cotwist of the 2-categorical adjunction $({\mathcal {O}}_{\Gamma ^t},{\mathcal {O}}_{\Gamma }, \eta,\varepsilon )$. Morphism $\eta$ in (23) is induced by the morphism ${\mathcal {O}}_{X}\to {\mathcal {O}}_D$ with kernel ${\mathcal {I}}_D$. Hence,

\[ {\rm diag}_*^X {\mathcal{I}}_D \to {\rm diag}_*^X{\mathcal{O}}_{X} \xrightarrow{\eta} {\rm diag}_*^X{\mathcal{O}}_D \to {\rm diag}_*^X {\mathcal{I}}_D[1] \]

is an exact triangle in ${\mathcal {D}}_{{\rm qc}}(X\times X)$. Thus, the cotwist equals ${\rm diag}_*^X{\mathcal {I}}_D$. It is an equivalence whose inverse is ${\rm diag}_*^X{\mathcal {I}}_D^{-1}$. The corresponding functorial exact triangle is (26).

Morphism $\varepsilon$ in (25) is induced by the truncation to the zeroth cohomology of the object $(i\times \mathop {{\rm Id}}\nolimits _D)^*(i\times \mathop {{\rm Id}}\nolimits _D)_*{\mathcal {O}}_{\Delta ^D}$. As we noted before, $(i\times \mathop {{\rm Id}}\nolimits _D)^*(i\times \mathop {{\rm Id}}\nolimits _D)_*{\mathcal {O}}_{\Delta ^D}$ has two non-zero cohomology sheaves and the truncation to the zeroth cohomology yields an exact triangle

\[ {\mathcal{O}}_{\Delta^D}\otimes {\mathcal{I}}_{D\times D}|_{D\times D}[1] \to (i\times\mathop{{\rm Id}}\nolimits_D)^*(i\times\mathop{{\rm Id}}\nolimits_D)_*{\mathcal{O}}_{\Delta^D} \to {\mathcal{O}}_{\Delta^D}\to {\mathcal{O}}_{\Delta^D}\otimes {\mathcal{I}}_{D\times D}|_{D\times D}[2]. \]

The twist

\[ {\mathcal {O}}_{\Delta ^D} \otimes {\mathcal {I}}_{D\times D}|_{D\times D}[1] \simeq {\rm diag}^D_*{\mathcal {I}}_D|_D[1] \]

is an equivalence whose inverse is ${\rm diag}^D_*{\mathcal {I}}_D^{-1}|_D[-1]$. The corresponding functorial exact triangle is (27).

3.2 $L^1f^*$ vanishing and consequences

Vanishing of $L^1f^*f_*(-)$ is local on $Y$, therefore throughout this section we assume that $Y$ is affine and fix a closed embedding $i\colon Y \to {\mathcal {Y}}$ of $Y$ as the zero locus of a regular function on a smooth affine ${\mathcal {Y}}$. As $Y\subset {\mathcal {Y}}$ is a principal Cartier divisor, the sheaf ${\mathcal {I}}_Y$ is isomorphic to ${\mathcal {O}}_{{\mathcal {Y}}}$.

The following lemma is one of the key technical points of this paper.

Lemma 3.2 Let $f\colon X\to Y$ satisfy assumption (p) and let $Y = \mathop {{\rm Spec}}\nolimits R$ be a spectrum of a complete Noetherian local ring with hypersurface singularities. Let further $D_i \subset X$ be as in Remark 2.7. Then $L^1f^*f_* {\mathcal {O}}_{D_i} \simeq 0$.

Proof. Let $\iota _i \colon D_i \to X$ denote the embedding of $D_i$ into $X$. We denote by $h$ the composite $h=if\iota _i \colon D_i \to \mathcal {Y}$. Morphism $h$ is finite, hence $h_*$ is exact and, by Grothendieck duality (A.1), we have

(28)\begin{equation} {\mathcal{E}} xt^i_{{\mathcal{Y}}}(Rh_* {\mathcal{O}}_{D_i}, {\mathcal{O}}_{{\mathcal{Y}}}) \simeq Rh_* {\mathcal{E}} xt^i_{D_i}({\mathcal{O}}_{D_i}, h^!{\mathcal{O}}_{{\mathcal{Y}}}) \simeq Rh_* {\mathcal{H}}_{D_i}^i(h^! {\mathcal{O}}_{{\mathcal{Y}}}). \end{equation}

Divisor $D_i$ is Cartier in a Gorenstein scheme, hence it is Gorenstein itself. As ${\mathcal {Y}}$ is smooth and it is the spectrum of a complete ring, we have $\omega _{{\mathcal {Y}}}^{\boldsymbol {\cdot }} \simeq {\mathcal {O}}_{{\mathcal {Y}}}[n+1]$. It follows that $h^! {\mathcal {O}}_{{\mathcal {Y}}}=h^!\omega _{{\mathcal {Y}}}^{\boldsymbol {\cdot }}[-n-1]=\omega _{D_i}[-2]$ is a sheaf in homological degree $2$. As $h_*$ is exact, $Rh_* {\mathcal {H}}_{D_i}^i(h^! {\mathcal {O}}_{{\mathcal {Y}}}) \simeq h_* {\mathcal {H}}_{D_i}^i(h^! {\mathcal {O}}_{{\mathcal {Y}}})$. Hence, by (28), we have

\[ {\mathcal{E}} xt^i_{\mathcal{Y}}(h_* {\mathcal{O}}_{D_i}, {\mathcal{O}}_{\mathcal{Y}}) = 0, \quad {\rm for}\ i\neq 2. \]

As $\mathcal {Y}$ is affine, $\mathop {{\rm Ext}}\nolimits _{{\mathcal {Y}}}^i(h_*{\mathcal {O}}_{D_i}, {\mathcal {O}}_{\mathcal {Y}}) = 0$, for $i\neq 2$. Hence, the projective dimension of $h_* {\mathcal {O}}_{D_i}$ as an ${\mathcal {O}}_{{\mathcal {Y}}}$ module is two. In other words, $h_* {\mathcal {O}}_{D_i}$ has a locally free resolution

\[ 0 \to {\mathcal{E}}_{-2} \to {\mathcal{E}}_{-1} \to {\mathcal{E}}_0 \to h_* {\mathcal{O}}_{D_i} \to 0. \]

Denote by ${\mathcal {E}}_{{\boldsymbol {\cdot }}}$ the complex $0 \to {\mathcal {E}}_{-2} \to {\mathcal {E}}_{-1} \to {\mathcal {E}}_0 \to 0$. The cohomology of complex $i_*i^* {\mathcal {E}}_{{\boldsymbol {\cdot }}}$ is ${\rm Tor}^{{\mathcal {Y}}}(h_* {\mathcal {O}}_{D_i},{\mathcal {O}}_Y)$. As $Y \subset \mathcal {Y}$ is a Cartier divisor, ${\mathcal {O}}_Y$ has a resolution of length two on ${\mathcal {Y}}$, i.e. the zeroth and first Tor groups only can be non-zero. As functor $i_*$ is exact, we conclude that the morphism $i^*{\mathcal {E}}_{-2} \to i^* {\mathcal {E}}_{-1}$ is injective.

Morphism $f$ is birational, hence the higher derived functors of $f^*$ are torsion. In particular, the kernel of $f^*i^* {\mathcal {E}}_{-2} \to f^*i^* {\mathcal {E}}_{-1}$ is torsion, meaning zero because $f^*i^* {\mathcal {E}}_{-2}$ is locally free. This shows that morphism $f^*i^* {\mathcal {E}}_{-2} \to f^*i^* {\mathcal {E}}_{-1}$ is also injective, thus complex $\widetilde {{\mathcal {E}}}_{\boldsymbol {\cdot }} := f^*i^* {\mathcal {E}}_{{\boldsymbol {\cdot }}}$ has zero cohomology except for ${\mathcal {H}}^{0}_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}})$ and ${\mathcal {H}}^{-1}_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}})$.

Let $E\subset X$ be the exceptional locus of $f$. Assume $L^1f^*f_* {\mathcal {O}}_{D_i} \neq 0$. Its support is contained in $E$. Moreover, because $L^1f^*f_* {\mathcal {O}}_{D_i} \in \mathscr {A}_f$ (see Lemma 2.9), the support of this sheaf should contain at least one component of the curve in the fiber over the closed point of $Y$ (see Theorem D.1 for the structure of the fiber). Therefore, there exists a closed point $x \in E \setminus D_i$ in the support of $L^1f^*f_* {\mathcal {O}}_{D_i}$.

By Theorem 3.1 for divisor $Y$ in ${\mathcal {Y}}$, object $i^* {\mathcal {E}}_{{\boldsymbol {\cdot }}} \simeq Li^*i_*f_* {\mathcal {O}}_{D_i}$ fits into an exact triangle

\[ f_* {\mathcal{O}}_{D_i}[1] \to i^* {\mathcal{E}}_{{\boldsymbol{\cdot}}} \to f_* {\mathcal{O}}_{D_i} \to f_* {\mathcal{O}}_{D_i}[2]. \]

By applying functor $L f^*$ to this triangle, we obtain an exact sequence

\[ f^* f_* {\mathcal{O}}_{D_i} \to \mathcal{H}_X^{-1}(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}}) \to L^1f^*f_* {\mathcal{O}}_{D_i}\to 0. \]

Sheaf $f^*f_*{\mathcal {O}}_{D_i}$ is supported on the preimage of $f(D_i) \subset Y$, i.e. on the set $E \cup D_i$. Let us restrict our attention to an open affine neighbourhood $\widetilde {i} \colon \widetilde {X}\hookrightarrow X$ of point $x$. The support of both sheaves $f^*f_* {\mathcal {O}}_{D_i}$ and $L^1f^*f_*{\mathcal {O}}_{D_i}$ is contained in $E$. Therefore, the assumptions on the exceptional locus for $f$ imply that ${\mathcal {H}}_X^{-1}(\widetilde {{\mathcal {E}}})$ is a non-zero sheaf with support of some codimension $l>1$.

Let $j\colon \widetilde {X}\to \mathcal {X}$ be a closed embedding into a smooth variety of dimension $m$. We have $\omega _{\widetilde {X}} \simeq \widetilde {i}^!f^! {\mathcal {O}}_Y \simeq \widetilde {i}^! {\mathcal {O}}_X \simeq {\mathcal {O}}_{\widetilde {X}}$, hence Grothendieck duality (A.1) gives

\[ {\mathcal{E}} xt^{{\boldsymbol{\cdot}}}_{\mathcal{X}}(j_* \widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}}, \omega_{\mathcal{X}}) \simeq j_* {\mathcal{E}} xt^{{\boldsymbol{\cdot}}}_{\widetilde{X}}(\widetilde{{\mathcal{E}}}_{\boldsymbol{\cdot}},{\mathcal{O}}_{\widetilde{X}}[n-m]). \]

Complex $\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}}$ consists of three locally free sheaves, thus ${\mathcal {E}} xt^k_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}}, {\mathcal {O}}_X)$ is zero, for $k>2$. It follows that ${\mathcal {E}} xt^k_{\mathcal {X}}(j_* \widetilde {{\mathcal {E}}}_{\boldsymbol {\cdot }}, \omega _{\mathcal {X}}) = 0$, for $k>m-n+2$. By applying functor ${\mathcal {H}} om^{{\boldsymbol {\cdot }}}_{\mathcal {X}}(-, \omega _{\mathcal {X}})$ to the exact triangle

\[ j_*{\mathcal{H}}^{-1}_X(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}})[1] \to j_*\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}} \to j_*{\mathcal{H}}^0_X(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}}) \to j_*{\mathcal{H}}^{-1}_X(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}})[2], \]

we obtain an isomorphism

(29)\begin{equation} {\mathcal{E}} xt^k_{\mathcal{X}}(j_*{\mathcal{H}}^{-1}_X(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}}), \omega_{\mathcal{X}}) \xrightarrow{\simeq} {\mathcal{E}} xt^{k+2}_{\mathcal{X}}(j_*{\mathcal{H}}^0_X(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}}), \omega_{\mathcal{X}}), \end{equation}

for any $k\geqslant m-n+2$. Sheaf $j_* {\mathcal {H}}^{-1}_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}})$ is non-zero with support of codimension $l+m-n \geqslant m-n+2$. It follows that sheaf ${\mathcal {E}} xt^{l+m-n}_{\mathcal {X}}(j_*{\mathcal {H}}^{-1}_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}}), \omega _{\mathcal {X}})$ is non-zero with support of codimension $l+m-n$, cf. [HL10, Proposition 1.1.6].

On the other hand, the codimension of the support of ${\mathcal {E}} xt^{l+m-n+2}_{\mathcal {X}}(j_*{\mathcal {H}}^0_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}}), \omega _{\mathcal {X}})$ is greater than or equal to $l+m-n+2$, which contradicts (29).

Lemma 3.3 Let $f\colon X \to Y$ satisfy assumption (d) and let $Y = \mathop {{\rm Spec}}\nolimits R$ be a spectrum of a complete Noetherian local ring. Let further ${\mathcal {M}}_i$ be the projective object in $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ as in (17) and ${\mathcal {N}}_i = {\mathcal {H}} om_X({\mathcal {M}}_i, {\mathcal {O}}_X)$ the projective object in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$. Then there exist exact sequences

(30)\begin{align} & 0 \to {\mathcal{O}}_X^{r_i} \to {\mathcal{M}}_i \to {\mathcal{O}}_{D_i} \to 0, \end{align}

(31)\begin{align} & 0 \to {\mathcal{N}}_i \to {\mathcal{O}}_X^{r_i} \to {\mathcal{O}}_{D_i} \to 0. \end{align}

Proof. The sheaf ${\mathcal {M}}_i$ is defined as an extension

\[ 0 \to {\mathcal{O}}_X^{r_i-1} \to {\mathcal{M}}_i \xrightarrow{q} {\mathcal{O}}_X(D_i) \to 0. \]

For any divisor $D_i$, one can find a linearly equivalent divisor $D'_i$ such that $D_i \cap D'_i$ is empty. Indeed, let $D'_i$ be a divisor as in Remark 2.7 linearly equivalent to $D_i$ which intersects $C_i$ in a point $x'_i$, not equal to $x_i = D_i \cap C_i$. Then $D_i\cap D'_i$ contains no closed points (as all closed points of $X$ lie on $C$), hence empty. It follows that ${\mathcal {O}}_{D_i}(D_i) \simeq {\mathcal {O}}_{D_i}(D'_i) \simeq {\mathcal {O}}_{D_i}$ and sequence

\[ 0 \to {\mathcal{O}}_X \to {\mathcal{O}}_X(D_i) \xrightarrow{r} {\mathcal{O}}_{D_i} \to 0 \]

is exact. As $\mathop {{\rm Ext}}\nolimits ^1_X({\mathcal {O}}_X, {\mathcal {O}}_X) \simeq 0$, the snake lemma for the two top rows of the following diagram gives the following commutative diagram in which rows and columns are exact sequences.

It gives exact sequence (30). By considering local homomorphisms into the structure sheaf we obtain sequence (31).

Lemma 3.4 Let $f\colon X\to Y$ satisfy assumption (p) and assume that $Y$ has hypersurface singularities. Then the sheaf $L^1f^*f_* {\mathcal {M}}$ is zero, for any ${\mathcal {M}}\in \mathscr {P}_{-1}$.

Proof. The statement is local in $Y$, therefore we might assume that $Y = \mathop {{\rm Spec}}\nolimits R$ is a spectrum of a complete Noetherian local ring with hypersurface singularities. Then ${\mathcal {M}}$ is a direct sum of copies of ${\mathcal {M}}_i$ and ${\mathcal {O}}_X$. Thus, it suffices to verify the lemma for ${\mathcal {M}}_i$, as clearly $L^1f^*{\mathcal {O}}_Y = 0$.

As $R^1f_* {\mathcal {O}}_X \simeq 0$, applying $Rf_*$ to sequence (30) gives an exact sequence

\[ 0 \to {\mathcal{O}}_Y^{r_i} \to f_* {\mathcal{M}}_i \to f_* {\mathcal{O}}_{D_i} \to 0. \]

It implies an exact sequence

\[ 0 \to L^1f^*f_* {\mathcal{M}}_i \to L^1f^*f_* {\mathcal{O}}_{D_i} \to {\mathcal{O}}_X^{r_i}. \]

By Lemma 3.2, the sheaf $L^1f^*f_* {\mathcal {O}}_{D_i}$ is zero, hence the result.

Lemma 3.5 Let $f \colon X\to Y$ satisfy assumption (p) and let $Y$ have hypersurface singularities. Let ${\mathcal {N}}$ be a locally projective object in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$. Then $f^*f_* {\mathcal {N}}$ is torsion-free and sequence

(32)\begin{equation} 0 \to f^*f_* {\mathcal{N}} \to {\mathcal{N}} \to \mathcal{Q} \to 0, \end{equation}

is exact for the sheaf $\mathcal {Q} ={\mathcal {H}}^0_X(\iota _{f*}\iota _f^*{\mathcal {N}})$, which is locally projective in $\mathscr {A}_f$.

Proof. First, we show that $f^*f_* {\mathcal {N}}$ is torsion-free. This can be done locally, i.e. we may assume that $Y$ is a spectrum of a complete Noetherian local ring. By [VdB04, Lemma 3.5.2] sheaf ${\mathcal {N}}$ is a direct sum of copies of ${\mathcal {O}}_X$ and ${\mathcal {N}}_i$, where ${\mathcal {N}}_i$ are vector bundles dual to ${\mathcal {M}}_i$.

By [VdB04, Lemma 3.1.2], the sheaf $R^1f_* {\mathcal {N}}_i$ is zero. By applying $f_*$ to sequence (31), we obtain an exact sequence on $Y$

(33)\begin{equation} 0 \to f_* {\mathcal{N}}_i \to {\mathcal{O}}_Y^{r_i} \to f_* {\mathcal{O}}_{D_i} \to 0. \end{equation}

As $f_*{\mathcal {O}}_{D_i}$ is a torsion sheaf, $f_*{\mathcal {N}}_i$ is non-zero. By applying $f^*$ to sequence (33), we obtain

\[ 0 \to L^1f^*f_* {\mathcal{O}}_{D_i} \to f^*f_* {\mathcal{N}}_i \to {\mathcal{O}}_X^{r_i}. \]

It implies that the torsion of $f^*f_* {\mathcal {N}}_i$ is $L^1f^*f_* {\mathcal {O}}_{D_i}$, which is zero by Lemma 3.2. Hence, sheaf $f^*f_* {\mathcal {N}}_i$ is torsion free and the counit morphism $\varepsilon _{{\mathcal {N}}}\colon f^*f_* {\mathcal {N}}_i \to {\mathcal {N}}_i$ is an embedding ($\varepsilon _{{\mathcal {N}}}$ is non-zero as under the $f^*\dashv f_*$ adjunction it corresponds to the identity endomorphism of $f_*{\mathcal {N}}_i$).

The cokernel of the morphism $f^*f_* {\mathcal {N}} \to {\mathcal {N}}$ is, by the definition of $\iota _f^*$, isomorphic to ${\mathcal {H}}_X^0(\iota _f^* {\mathcal {N}})$, hence $\mathcal {Q}$ is an object in $\mathscr {A}_f$. Now let $E$ be any object in $\mathscr {A}_f$. Then $E$, considered as an object in ${\mathcal {D}}^b(X)$, lies in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$, hence $\mathop {{\rm Ext}}\nolimits ^1_X({\mathcal {N}}, E) \simeq 0$. Moreover, $f^* \vdash f_*$ adjunction implies that $\mathop {{\rm Hom}}\nolimits _X(f^*f_* {\mathcal {N}}, E) \simeq 0$. Thus, by applying higher derived functors of $\mathop {{\rm Hom}}\nolimits _X(-,E)$ to sequence (32), we obtain $\mathop {{\rm Ext}}\nolimits ^1_X(\mathcal {Q}, E) \simeq 0$, which proves that $\mathcal {Q} \in \mathscr {A}_f$ is indeed a projective object.

Remark 3.6 In the complete local case, ${\mathcal {P}}$ in formula (12) is isomorphic to $\mathcal {Q}$ of Lemma 3.5. Indeed, applying $\iota _f^*$ to sequences (30) and (31) and using the fact that $\iota _f^*({\mathcal {O}}_X) \simeq 0$, we obtain an isomorphism ${\mathcal {H}}_X^0(\iota _f^* {\mathcal {N}}_i) \simeq {\mathcal {H}}_X^{-1}(\iota _f^* {\mathcal {O}}_{D_i}) \simeq {\mathcal {H}}^{-1}_X(\iota _f^* {\mathcal {M}}_i)$.

The $t$-structure on ${\mathcal {D}}^b(X)$ with heart $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ is obtained from the standard $t$-structure by a tilt in the torsion pair $({\mathcal {T}}_{-1}, {\mathcal {F}}_{-1})$ as in (10) and (11), see [VdB04, Lemma 3.1.2]. Hence, for any $F \in {\mathcal {D}}^b(X)$, sequence

(34)\begin{equation} 0 \to {\mathcal{F}}_{-1}({\mathcal{H}}_X^{i-1}(F))[1] \to {\mathcal{H}}_{{}^{-1}{\rm Per}}^i(F) \to {\mathcal{T}}_{-1}({\mathcal{H}}^i_X(F)) \to 0 \end{equation}

is exact in $\mathop {{\rm Coh}}\nolimits (X)$, where by ${\mathcal {T}}_{-1}(G)$ and ${\mathcal {F}}_{-1}(G)$ we denote the torsion and torsion-free part, respectively, of a coherent sheaf $G$ with respect to the torsion pair $({\mathcal {T}}_{-1}, {\mathcal {F}}_{-1})$.

Proposition 3.7 Let $f\colon X\to Y$ satisfy assumption (p) and assume that $Y$ has hypersurface singularities. Let ${\mathcal {M}}$ be a projective object in ${}^{-1}{\rm Per}(X/Y)$. Then

\[ {\mathcal{H}}^0_{{}^{-1}{\rm Per}}(Lf^*f_* {\mathcal{M}}) = f^*f_* {\mathcal{M}}. \]

Proof. Lemma 3.8 implies that $f^*f_* {\mathcal {M}}$ is an object in $\mathop {{\rm Coh}}\nolimits (X) \cap {}^{-1}{\rm Per}(X/Y)$, hence ${\mathcal {T}}_{-1}(f^*f_* {\mathcal {M}}) = f^*f_* {\mathcal {M}}$. Moreover, $L^1f^*f_* {\mathcal {M}} = 0$ by Lemma 3.4. We conclude by sequence (34).

Lemma 3.8 Let $f \colon X\to Y$ satisfy assumption (d) and let $E$ be a coherent sheaf on $Y$. Then $f^*E$ is an object in ${}^{p}{\rm Per}(X/Y)$, for $p = -1,0$. If, moreover, $E$ is locally free and $Y$ is affine, then $f^* E$ is an object in $\mathscr {P}_p$, for $p=-1,0$.

Proof. We have ${\rm Coh}(X) \cap {}^{0}{\rm Per}(X/Y) = {\mathcal {T}}_{0}$ and ${\rm Coh}(X) \cap {}^{-1}{\rm Per}(X/Y) = {\mathcal {T}}_{-1}$, for categories ${\mathcal {T}}_0$ and ${\mathcal {T}}_{-1}$ defined in (8) and (10), respectively.

Let $E$ be a coherent sheaf on $Y$. By Lemma 2.9, sheaf $R^1f_*f^* E$ is zero, hence $f^*E$ is an object in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$. From adjunction

\[ \mathop{{\rm Hom}}\nolimits_X(f^* E, {\mathcal{A}}_f) \simeq \mathop{{\rm Hom}}\nolimits_Y(E, f_* {\mathcal{A}}_f) \simeq 0, \]

it follows that $f^*E$ lies also in $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$.

Now we assume further that $E$ is locally free and $Y$ is affine. Let $E'$ be any object in $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$, for $p=-1, 0$. Then

\[ \mathop{{\rm Ext}}\nolimits^1_X(f^*E, E') \simeq \mathop{{\rm Ext}}\nolimits^1_X(Lf^* E, E') \simeq \mathop{{\rm Ext}}\nolimits^1_Y(E, Rf_* E'). \]

Because $Rf_*$ is exact for the perverse $t$-structures on ${\mathcal {D}}^b(X)$, object $Rf_* E'$ is a coherent sheaf on $Y$. As $E$ is locally free and $Y$ is affine, $\mathop {{\rm Ext}}\nolimits ^1_Y(E, Rf_* E') \simeq 0$, i.e. $f^*E$ is projective in $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$.

Remark 3.9 For any locally free sheaf ${\mathcal {E}}$ on $Y$, we also have an isomorphism ${\mathcal {H}}^0_{\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)}(Lf^*{\mathcal {E}}) = f^*{\mathcal {E}}$. Indeed, $Lf^* {\mathcal {E}}$ is isomorphic to $f^* {\mathcal {E}}$ and the latter is an object in $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$, by Lemma 3.8.

3.3 The equivalence of reflexive sheaves under flopping contractions

Recall that a sheaf $F$ is normal when, for any open sets $U \subset V$ such that codimension of $V\setminus U$ in $V$ is greater than one, the restriction morphism $F(V) \to F(U)$ is an isomorphism.

Lemma 3.10 [Har80, Proposition 1.6]

Let $X$ be a normal, integral Noetherian scheme. A sheaf $F$ on $X$ is reflexive if and only if it is torsion-free and normal.

For a scheme $X$, denote by $\textrm{Ref}\,(X)$ the category of reflexive sheaves.

Lemma 3.11 (Cf. [VdB04, Lemma 4.2.1])

Let $f \colon X\to Y$ be a projective birational morphism, $X,Y$ normal Noetherian and the exceptional locus of $f$ have codimension $>1$ in $X$. Then the following functors are mutually inverse equivalences:

\[ f_*\colon {\rm Ref}(X) \to {\rm Ref}(Y),\quad (f^*(-))^{\vee \vee} \colon {\rm Ref}(Y) \to {\rm Ref}(X). \]

Proof. For a torsion-free sheaf $F$ on $X$, its push-forward $f_* F$ is also torsion-free. Thus, $f_* F$ will be reflexive if we check that it is normal.

Let $G$ be a torsion-free sheaf on $Y$ which is not normal. There exist open sets $j \colon U \hookrightarrow V$ with the complement of $U$ in codimension greater than one in $V$ such that canonical morphism $G \to j_*j^* G$ is not an isomorphism. Then we have a non-trivial extension

\[ 0 \to G \to j_*j^*G \to Q \to 0. \]

The question is local, so we can assume $Y$ to be affine. Sheaf $Q$ is supported in codimension greater than one, hence there exists a closed subscheme $Z \subset Y$ of codimension greater than one such that sequence

\[ 0 \to K \to {\mathcal{O}}_{Z}^{\oplus k} \to Q \to 0 \]

is exact. As $G$ is torsion-free, $\mathop {{\rm Hom}}\nolimits (K, G)$ is zero. Hence, vanishing of $\mathop {{\rm Ext}}\nolimits ^1_Y({\mathcal {O}}_Z, G)$ implies that $\mathop {{\rm Ext}}\nolimits ^1_Y(Q, G)$ is also zero. Thus, in order to show that a torsion-free sheaf $G$ is normal it suffices to check that $\mathop {{\rm Ext}}\nolimits ^1_Y({\mathcal {O}}_Z, G)$ vanishes, for any closed $Z \subset Y$ of codimension greater than one.

Group $\mathop {{\rm Ext}}\nolimits ^1_Y({\mathcal {O}}_Z, f_* F)$ is zero if and only if morphism $\mathop {{\rm Hom}}\nolimits ({\mathcal {O}}_Y, f_* F) \to \mathop {{\rm Hom}}\nolimits (I_Z, f_* F)$ given by short exact sequence

\[ 0 \to I_Z \to {\mathcal{O}}_Y \to {\mathcal{O}}_Z \to 0 \]

is an isomorphism. By adjunction, $\mathop {{\rm Hom}}\nolimits _Y(I_Z, f_* F)$ is isomorphic to $\mathop {{\rm Hom}}\nolimits _X(f^* I_Z, F)$. Exceptional locus of $f$ has codimension greater than one, hence $f^* I_Z$ is isomorphic to ${\mathcal {O}}_X$ in codimension one. As $F$ is reflexive, this implies $\mathop {{\rm Hom}}\nolimits _X(f^*I_Z, F)= \mathop {{\rm Hom}}\nolimits _X({\mathcal {O}}_X, F)$.

Thus, functor $f_*$ maps reflexive sheaves on $X$ to reflexive sheaves on $Y$. Clearly, the same is true about $(f^*(-))^{\vee \vee }$.

Now let $F$ be an object in $\textrm{Ref}\,(X)$. Adjunction implies a morphism $\alpha \colon (f^*f_*F)^{\vee \vee } \to F^{\vee \vee } \simeq F$. Both sheaves are reflexive and $\alpha$ is an isomorphism outside exceptional locus of $f$, which has codimension greater than one. Therefore, $\alpha$ is an isomorphism.

For $G \in \textrm{Ref}\,(Y)$, reflexification $f^*G \to (f^*G)^{\vee \vee }$ together with adjunction $G\to f_*f^*G$ give $\beta \colon G \to f_*((f^*G)^{\vee \vee })$. Again, morphism $\beta$ is an isomorphism in codimension one between reflexive sheaves, hence an isomorphism.

Finally, a morphism between reflexive sheaves is determined by its restriction to any open set with the complement of codimension greater than one. Thus, functors $f_*$ and $(f^*(-))^{\vee \vee }$ give isomorphisms on the groups of morphisms between objects in $\textrm{Ref}\,(X)$ and $\textrm{Ref}\,(Y)$.

Let $f\colon X\to Y$ satisfy assumption (a). For ${\mathcal {M}}$ an object in $\mathscr {P}_{-1}$, put

(35)\begin{equation} {\mathcal{N}}^+ := (f^{+*}f_* {\mathcal{M}})^{\vee \vee}. \end{equation}

By Lemma 3.11, sheaf ${\mathcal {N}}^+$ belongs to $\textrm{Ref}\,(X^+)$.

Proposition 3.12 Let $f\colon X\to Y$ satisfy assumption (a) and $f^+ \colon X^+ \to Y$ its flop. Then functor

\[ (f^{+*}f_*(-))^{\vee \vee} \colon {\rm Ref}(X) \to {\rm Ref}(X^+) \]

restricts to equivalences $\mathscr {P}_{-1} \xrightarrow {\simeq } \mathscr {P}_0^+$ and $\mathscr {P}_0 \xrightarrow {\simeq } \mathscr {P}_{-1}^+$.

Proof. By [VdB04, Proposition 3.2.6], categories $\mathscr {P}_{-1}$ and $\mathscr {P}_0$ are subcategories of $\textrm{Ref}\,(X)$. By [VdB04, Proposition 4.3.1], subcategories $f_* \mathscr {P}_{-1}$ and $f^+_* \mathscr {P}_0^+$ of $\textrm{Ref}\,(Y)$ are equivalent. By Lemma 3.11, the subcategory $(f^{+*}f_* \mathscr {P}_{-1})^{\vee \vee }$ of $\textrm{Ref}\,(X^+)$ corresponds to $\mathscr {P}_{-1}$ under equivalences $\textrm{Ref}\,(X)\simeq \textrm{Ref}\,(Y) \simeq \textrm{Ref}\,(X^+)$, hence $\mathscr {P}_0^+ \simeq (f^{+*}f_* \mathscr {P}_{-1})^{\vee \vee }$.

Exchanging the roles of $f$ and $f^+$, we get equivalence of $\mathscr {P}_{-1}^+$ with $\mathscr {P}_0$.

3.4 A divisorial embedding into a smooth scheme and cohomology

We assume that $Y$ is affine and fix a closed embedding $i\colon Y \to {\mathcal {Y}}$ of $Y$ as the zero locus of a regular function on a smooth affine ${\mathcal {Y}}$. Denote $g= i \circ f$, i.e. consider the following commutative diagram.

We describe objects $Lg^*g_* {\mathcal {M}}$ and $Lg^*f_*{\mathcal {N}}$, for any ${\mathcal {M}}\in \mathscr {P}_{-1}$, ${\mathcal {N}} \in \mathscr {P}_0$. First, we prove that the sheaves $g_* {\mathcal {M}}$, $g_*{\mathcal {N}}$ have short locally free resolutions.

Recall that a morphism $f\colon X\to Y$ is crepant if $Lf^*(\omega _Y^{\boldsymbol {\cdot }})\simeq \omega _X^{\boldsymbol {\cdot }}.$

Lemma 3.13 A proper morphism $f\colon X\to Y$ of Gorenstein varieties is crepant if and only if $f^!({\mathcal {O}}_Y) \simeq {\mathcal {O}}_X$.

Proof. As $X$ and $Y$ are Gorenstein, $\omega _X^{\boldsymbol {\cdot }}$ and $\omega _Y^{\boldsymbol {\cdot }}$ are, up to shift, line bundles. In particular, $\omega _Y^{\boldsymbol {\cdot }}$ is a perfect complex, hence, by Lemma A.1, $f^!(\omega _Y^{\boldsymbol {\cdot }}) \simeq f^!({\mathcal {O}}_Y) \otimes Lf^*(\omega _Y^{\boldsymbol {\cdot }})$. Isomorphism $\omega _X^{\boldsymbol {\cdot }} \simeq f^!(\omega _Y^{\boldsymbol {\cdot }}) \simeq f^!({\mathcal {O}}_Y)\otimes Lf^*(\omega _Y)^{\boldsymbol {\cdot }}$ implies $f^!({\mathcal {O}}_Y) \simeq \omega _X^{\boldsymbol {\cdot }} \otimes (Lf^*(\omega _Y^{\boldsymbol {\cdot }}))^{-1}$. Hence, $Lf^*(\omega _Y^{\boldsymbol {\cdot }}) \simeq \omega _X^{\boldsymbol {\cdot }}$ if and only $f^!({\mathcal {O}}_Y) \simeq {\mathcal {O}}_X$.

Lemma 3.14 Let $X$, $Y$ be Gorenstein and $f\colon X\to Y$ a crepant morphism satisfying assumption (d). Assume $Y$ is affine and $i\colon Y \to {\mathcal {Y}}$ an embedding of $Y$ as a Cartier divisor into a smooth ${\mathcal {Y}}$. Denote $g = i\circ f\colon X\to {\mathcal {Y}}$. Then $g_* {\mathcal {M}}$ and $g_* {\mathcal {N}}$, for ${\mathcal {M}} \in \mathscr {P}_{-1}$, ${\mathcal {N}} \in \mathscr {P}_0$, admit locally free resolutions of length two.

Proof. Let ${\mathcal {M}}$ be an object in $\mathscr {P}_{-1}$. As ${\mathcal {Y}}$ is affine, the length of the locally free resolution for $g_* {\mathcal {M}}$ is $l+1$, for the maximal $l$ such that ${\mathcal {E}} xt^l_{{\mathcal {Y}}}(g_* {\mathcal {M}}, {\mathcal {O}}_{{\mathcal {Y}}})$ is non-zero. Thus, the question is local on ${\mathcal {Y}}$ and, by restricting to a smaller affine open subset, we can assume that $\omega ^{{\boldsymbol {\cdot }}}_{{\mathcal {Y}}} \simeq {\mathcal {O}}_{{\mathcal {Y}}}[n+1]$ and $\omega ^{{\boldsymbol {\cdot }}}_{Y} \simeq {\mathcal {O}}_{Y}[n]$. Then $\omega _X^{{\boldsymbol {\cdot }}} \simeq {\mathcal {O}}_X[n]$, because $f$ is crepant. Let ${\mathcal {N}} = R {\mathcal {H}} om_X({\mathcal {M}}, {\mathcal {O}}_X)={\mathcal {H}} om_X({\mathcal {M}}, {\mathcal {O}}_X)$. It is an object in $\mathscr {P}_0$, see [VdB04, Proposition 3.2.6]. As ${\mathcal {N}}$ is an object in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$, we have $Rg_* {\mathcal {N}} \simeq g_* {\mathcal {N}}$. Then we have, by Grothendieck duality (A.1),

\[ R{\mathcal{H}} om_{{\mathcal{Y}}}(g_* {\mathcal{M}}, {\mathcal{O}}_{{\mathcal{Y}}})\simeq R{\mathcal{H}} om_{{\mathcal{Y}}}(Rg_* {\mathcal{M}}, \omega_{{\mathcal{Y}}})[-1] \simeq Rg_* R {\mathcal{H}} om_X({\mathcal{M}}, \omega_X)[-1]\simeq g_* {\mathcal{N}}[-1]. \]

Hence, $g_* {\mathcal {M}}$ is of projective dimension one. Analogously, for ${\mathcal {N}} \in \mathscr {P}_0$, its dual ${\mathcal {M}} = R{\mathcal {H}} om_X({\mathcal {N}},{\mathcal {O}}_X)$ lies in $\mathscr {P}_{-1}$ and $R{\mathcal {H}} om(g_* {\mathcal {N}}, {\mathcal {O}}_{{\mathcal {Y}}}) \simeq g_* {\mathcal {M}}[-1]$.

Proposition 3.15 Let $f\colon X\to Y$ be as in Lemma 3.14. Let further $p\in \{-1,0\}$ and ${\mathcal {R}} \in \mathscr {P}_p$ be a projective object in $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$. Then the object $Lg^*Rg_*{\mathcal {R}}$ has cohomology

\[ {\mathcal{H}}^i_X(Lg^*Rg_* {\mathcal{R}}) = \begin{cases}{\mathcal{R}}, & \textrm{if }\, i=-1,\\ f^*f_* {\mathcal{R}}, & \textrm{if } i=0, \\ 0, & \textrm{otherwise.} \end{cases} \]

The same cohomology is for the $t$-structure with heart $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$.

Proof. As $R^1g_*{\mathcal {R}} =0$, the zero cohomology of $Lg^*Rg_*{\mathcal {R}}$ is the sheaf $g^*g_*{\mathcal {R}}$. The adjunction counit for $i\colon Y \to {\mathcal {Y}}$ gives an isomorphism $g^*g_*\xrightarrow {\simeq } f^*f_*$, hence ${\mathcal {H}}^0_X(Lg^*Rg_*{\mathcal {R}}) = f^*f_*{\mathcal {R}}$.

As $g_*{\mathcal {R}}$ admits a locally free resolution (see Lemma 3.14):

\[ 0 \to {\mathcal{E}}_{-1} \to {\mathcal{E}}_0 \to g_* {\mathcal{R}} \to 0, \]

the sheaves ${\mathcal {H}}^i_X(Lg^*Rg_*{\mathcal {R}})$ are zero, for $i\leqslant -2$. Moreover, $L^1g^*g_*{\mathcal {R}}$ is reflexive, as the kernel of a morphism of locally free sheaves [Har80, Proposition 1.1].

Applying $Lf^*$ to exact triangle $f_* {\mathcal {R}}[1] \to Li^*g_* {\mathcal {R}} \to f_* {\mathcal {R}} \to f_*{\mathcal {R}}[2]$ (given by Theorem 3.1 for $Y\subset {\mathcal {Y}}$) implies the exact sequence

\[ L^2f^*f_*{\mathcal{R}}\to f^*f_*{\mathcal{R}}\to L^1g^*g_*{\mathcal{R}}\to L^1f^*f_*{\mathcal{R}}. \]

As $f$ is an isomorphism in codimension one, the sheaves $L^2f^*f_*{\mathcal {R}}$ and $L^1f^*f_*{\mathcal {R}}$ have the support of codimension at least two. As the universal map $f^*f_*{\mathcal {R}}\to (f^*f_*{\mathcal {R}})^{\vee \vee }$ is also an isomorphism in codimension one, the induced map $(f^*f_*{\mathcal {R}})^{\vee \vee } \to L^1g^*g_*{\mathcal {R}}$ is an isomorphism in codimension one. Hence, it is an isomorphism, see Lemma 3.10. It follows that $L^1g^*g_*{\mathcal {R}} \simeq (f^*f_*{\mathcal {R}})^{\vee \vee }$. As ${\mathcal {R}}$ is locally free [VdB04, Proposition 3.2.6], $(f^*f_*{\mathcal {R}})^{\vee \vee }\simeq {\mathcal {R}}$ by Lemma 3.11.

By Lemma 3.8, $f^*f_*{\mathcal {R}}$ is an object of $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$. As the same holds for ${\mathcal {R}} \in \mathscr {P}_{p} \subset \mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$, the exact triangle ${\mathcal {R}}[1] \to Lg^*g_*{\mathcal {R}} \to f^*f_*{\mathcal {R}}$ is also the decomposition of $Lg^*g_*{\mathcal {R}}$ in the $t$-structure with heart $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$.

Proposition 3.15 implies that triangle

(36)\begin{equation} {\mathcal{M}}[1] \to Lg^*g_* {\mathcal{M}} \to f^*f_* {\mathcal{M}} \to {\mathcal{M}}[2] \end{equation}

is exact. By Lemma 3.14, object $Lg^*g_* {\mathcal {M}}$ is quasi-isomorphic to a complex $f^* {\mathcal {F}}_{-1} \to f^* {\mathcal {F}}_0$, for some locally free sheaves ${\mathcal {F}}_{-1}$, ${\mathcal {F}}_0$ on $Y$. Hence, sequence

(37)\begin{equation} 0 \to {\mathcal{M}} \to f^* {\mathcal{F}}_{-1} \to f^* {\mathcal{F}}_0 \to f^*f_* {\mathcal{M}} \to 0 \end{equation}

is a projective resolution for $f^*f_* {\mathcal {M}}$ in $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$, see Lemma 3.8.

Note that short exact sequence (15) and Lemma 3.8 imply that

(38)\begin{equation} 0 \to f^*f_* {\mathcal{M}} \to {\mathcal{M}} \to {\mathcal{P}}[1] \to 0 \end{equation}

is a short exact sequence in ${}^{-1}{\rm Per}(X/Y)$.

3.5 2-periodicity

Proposition 3.16 Let $f \colon X\to Y$ satisfy assumption (d) and assume that $Y \subset {\mathcal {Y}}$ is a principal divisor in a smooth affine ${\mathcal {Y}}$. Then there exists a morphism $Lf^* \to Lf^*[2]$ of functors ${\mathcal {D}}_{{\rm qc}}(Y) \to {\mathcal {D}}_{{\rm qc}}(X)$, which, for any $E$ in ${\mathcal {D}}^b(Y)$, is an isomorphism on almost all cohomology sheaves of $Lf^*(E)$.

Proof. By applying $Lf^*$ to functorial exact triangle (27), we obtain a morphism $Lf^*E \to Lf^*E[2]$ with cone $Lf^*Li^*i_*E[1]$. As ${\mathcal {Y}}$ is smooth, ${\mathcal {D}}^b({\mathcal {Y}})$ coincides with the category of perfect complexes on ${\mathcal {Y}}$. It follows that the functor $Lf^*Li^*i_*$ takes ${\mathcal {D}}^b(Y)$ to ${\mathcal {D}}^b(X)$ (even to $\mathop {{\rm Perf}}\nolimits (X)$), hence, for any $E\in {\mathcal {D}}^b(Y)$, there exists $l\in \mathbb {Z}$ such that ${\mathcal {H}}^i(Lf^*Li^*i_*E) = 0$, for $i < l$. Then ${\mathcal {H}}^i Lf^*E \simeq {\mathcal {H}}^{i-2} Lf^* E$, for any $i< l$.

Now let $f\colon X\to Y$ be a crepant morphism of Gorenstein varieties satisfying assumption (p) and assume that $Y \subset {\mathcal {Y}}$ is a principal divisor in a smooth ${\mathcal {Y}}$. Then ${\mathcal {I}}_Y\simeq {\mathcal {O}}_{{\mathcal {Y}}}$, hence composing functorial exact triangle (27) with $Lf^*$ and precomposing with $Rf_*$ gives a functorial exact triangle

(39)\begin{equation} Lf^*Rf_*[1] \to Lg^*Rg_* \to Lf^*Rf_* \to Lf^*Rf_* [2]. \end{equation}

Object $Lg^*g_* {\mathcal {M}}$ has only two non-zero cohomology sheaves described by Proposition 3.15, for any ${\mathcal {M}}\in \mathscr {P}_{-1}$. Then the long exact sequence of cohomology sheaves for triangle (39) applied to ${\mathcal {M}}$ implies short exact sequence

\[ 0 \to L^2f^*f_* {\mathcal{M}} \to f^*f_* {\mathcal{M}} \to {\mathcal{M}} \to 0 \]

and isomorphisms $L^lf^*f_* {\mathcal {M}} \simeq L^{l+2} f^*f_* {\mathcal {M}}$, for $l \geqslant 1$. Hence, we have 2-periodicity

(40)\begin{equation} L^lf^*f_* {\mathcal{M}} \simeq \begin{cases}{\mathcal{P}}, & \text{for even }\, l \geqslant 2,\\ 0, & \text{for odd }\, l. \end{cases} \end{equation}

Remark 3.17 From Propositions 3.16 and 3.15 it follows that

\[ L^lf^*f_* {\mathcal{N}} \simeq \begin{cases} \mathcal{Q}, & \text{for odd }\, l\geqslant 1,\\ 0, & \text{for even }\,l\geqslant 2.\end{cases} \]

Remark 3.18 Let $f\colon X\to Y$ be a crepant morphism of Gorenstein varieties satisfying assumption (d) such that $Y \subset {\mathcal {Y}}$ is a principal divisor in a smooth affine ${\mathcal {Y}}$. Using Proposition 3.15 we construct a projective resolution

(41)\begin{equation} 0 \to {\mathcal{N}} \to f^* {\mathcal{E}}_{-1} \to f^* {\mathcal{E}}_0 \to f^*f_* {\mathcal{N}} \to 0 \end{equation}

of $f^*f_* {\mathcal {N}}$ in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$. Here, ${\mathcal {E}}_{-1}$ and ${\mathcal {E}}_0$ are locally free sheaves on $Y$ such that $Li^*g_* {\mathcal {N}}$ is quasi-isomorphic to ${\mathcal {E}}_{-1}\to {\mathcal {E}}_0$ (see Lemma 3.14).

Proposition 3.19 (Cf. [DW16, Proposition 5.6])

Let $f\colon X\to Y$ satisfy assumption (a), ${\mathcal {M}}$ be in $\mathscr {P}_{-1}$ and ${\mathcal {N}}$ in $\mathscr {P}_0$. Then there exist locally free sheaves ${\mathcal {F}}_{-1}$, ${\mathcal {F}}_0$, ${\mathcal {E}}_{-1}$, ${\mathcal {E}}_0$ on $Y$ such that

\begin{align*} &0 \to {\mathcal{M}} \to f^*{\mathcal{F}}_{-1} \to f^* {\mathcal{F}}_0 \to {\mathcal{M}} \to {\mathcal{P}}_{{\mathcal{M}}}[1] \to 0\\ &0 \to {\mathcal{N}} \to f^*{\mathcal{E}}_{-1} \to f^* {\mathcal{E}}_0 \to {\mathcal{N}} \to {\mathcal{P}}_{{\mathcal{N}}} \to 0 \end{align*}

are projective resolutions of ${\mathcal {P}}_{{\mathcal {M}}}[1] = {\mathcal {H}}^{-1}_X \iota _f^* {\mathcal {M}}[1]$ and ${\mathcal {P}}_{{\mathcal {N}}} = {\mathcal {H}}^0_X \iota _f^* {\mathcal {N}}$ respectively in ${}^{-1}{\rm Per}(X/Y)$ and $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$.

Proof. Composing resolution (37) with sequence (38) gives the projective resolution for ${\mathcal {P}}_{{\mathcal {M}}}[1]$ Similarly, resolution (41) and sequence (32) give projective resolution for ${\mathcal {P}}_{{\mathcal {N}}}$.

4. The flop functor and Van den Bergh's functor

Now let $X$, $X^+$ and $Y$ be quasi-projective Gorenstein varieties of dimension $n$ such that $X$ and $X^+$ are related by a flop over $Y$. We assume that $Y$ has canonical hypersurface singularities of multiplicity two. Note that this condition is satisfied if $Y$ has dimension three and terminal Gorenstein singularities. We assume that fibers of $f$ have dimension bounded by one and the exceptional locus of $f$ has codimension greater than one in $X$. Then morphism $f^+$ satisfies the same conditions.

Van den Bergh in [VdB04] proved an equivalence of ${\mathcal {D}}^b(X)$ with ${\mathcal {D}}^b(X ^+)$ under these assumptions.

We consider a diagram

(42)

and the flop functor

(43)\begin{equation} F= Rp^+_* Lp^*\colon {\mathcal{D}}_{{\rm qc}}(X) \to {\mathcal{D}}_{{\rm qc}}(X^+). \end{equation}

In this section we give an alternative description of the flop functor and the functor $\Sigma$ considered in [VdB04] under an extra assumption that $Y$ is affine. We show that these are inverse to each other. Keeping the assumption that $Y$ is affine we also provide an alternative description of the flop–flop functor

(44)\begin{equation} F^+F = Rp_*\, Lp^{+*}\, Rp^+_*\, Lp^*\colon {\mathcal{D}}_{{\rm qc}}(X) \to {\mathcal{D}}_{{\rm qc}}(X). \end{equation}

We construct a functorial exact triangle relating $\Sigma$, $F$ and the derived push-forward to a smooth scheme ${\mathcal {Y}}$. We conclude that the flop functor induces an equivalence ${\mathcal {D}}^b(X) \xrightarrow {\simeq }{\mathcal {D}}^b(X^+)$ and, following [Che02], show that $F$ is an equivalence also in the case when $Y$ is not affine.

Lemma 4.1 Let $g\colon X\to Z$ be a proper surjective morphism with connected fibers. If $Z$ is normal, then $g_* {\mathcal {O}}_X \simeq {\mathcal {O}}_Z$.

Proof. For Stein decomposition $g \colon X\to \mathop {{\rm Spec}}\nolimits _Z g_* {\mathcal {O}}_X \xrightarrow {\varphi } Z$, morphism $\varphi$ is finite and has connected fibers, hence it is birational. As a finite birational morphism onto a normal variety is an isomorphism, $\varphi$ is an isomorphism. It follows that $g_* {\mathcal {O}}_X \simeq {\mathcal {O}}_Z$.

Remark 4.2 We assume that $X$ and $X^+$ are normal varieties. As morphism $p$ is proper, surjective and with connected fibers, Lemma 4.1 implies that $p_* {\mathcal {O}}_{X\times _Y X^+}\simeq {\mathcal {O}}_X$. Moreover, by Proposition 2.11, the sheaf $R^1p_* {\mathcal {O}}_{X\times _Y X^+}$ vanishes. It follows that $p$ satisfies assumption (d).

4.1 An alternative description of Van den Bergh's functor

Lemma 4.3 Let $f\colon X\to Y$ satisfy assumption (d), $Y$ affine and ${\mathcal {M}}_1$, ${\mathcal {M}}_2$ objects in $\mathscr {P}_{p}$, for $p=-1$ or $0$. Then $\mathop {{\rm Ext}}\nolimits _X^i({\mathcal {M}}_1, {\mathcal {M}}_2) = 0$, for $i \geqslant 1$.

Proof. By [VdB04, Proposition 3.2.6], objects in $\mathscr {P}_{p}$ are vector bundles on $X$. Hence, ${\mathcal {E}} xt^i({\mathcal {M}}_1, {\mathcal {M}}_2) \simeq 0$, for $i > 0$. As morphism $f$ has fibers of dimension bounded by one, the local-to-global spectral sequence implies that $\mathop {{\rm Ext}}\nolimits ^i({\mathcal {M}}_1, {\mathcal {M}}_2)\simeq H^0(Y, R^if_* {\mathcal {H}} om({\mathcal {M}}_1, {\mathcal {M}}_2))$ vanishes, for $i> 1$. Group $\mathop {{\rm Ext}}\nolimits ^1_X({\mathcal {M}}_1,{\mathcal {M}}_2)$ is isomorphic to $\mathop {{\rm Ext}}\nolimits ^1_{\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)}({\mathcal {M}}_1, {\mathcal {M}}_2)$, hence also zero, as ${\mathcal {M}}_1$ is an object of $\mathscr {P}_p$.

Let $\mathscr {P} \subset {\mathcal {B}}$ be a full subcategory of an abelian category closed under extensions. Denote by ${\rm Hot}_{{\mathcal {B}}}^{-,b}(\mathscr {P})$ the homotopy category of bounded above complexes of objects in $\mathscr {P}$ with bounded cohomology in ${\mathcal {B}}$. We use the notation ${\rm Hot}^{-,b}(\mathscr {P})$ when the category ${\mathcal {B}}$ is clear from the context. Denote by ${\rm Hot}^{-,b}(\mathscr {P})^{\leqslant l}$ the full subcategory of complexes with non-zero cohomology in degree up to $l$. Denote by ${\rm Hot}^-(\mathscr {P})$ the category of bounded above complexes of objects in $\mathscr {P}$ without any constraint on cohomology. Finally, denote by ${\rm Hot}^b(\mathscr {P})$ the full subcategory of bounded complexes in ${\rm Hot}^{-,b}(\mathscr {P})$. Categories ${\rm Hot}^{-,b}(\mathscr {P})$, ${\rm Hot}^-(\mathscr {P})$ and ${\rm Hot}^b(\mathscr {P})$ are triangulated by [Nee90]. If $\mathscr {P}$ is the category of projective objects in $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$, for $p=-1,0$, then we use the above notation for the embedding $\mathscr {P}_p \subset \mathop {{\rm Coh}}\nolimits (X)$.

Assume that morphism $f\colon X\to Y$ satisfies assumption (a). As $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$ has enough projective objects, for $p=-1$ or $0$, category ${\mathcal {D}}^b(\mathop {{}^{p}{\rm Per}}\nolimits (X/Y))$ is equivalent to ${\rm Hot}^{-,b}(\mathscr {P}_{p})$, see [GM03, Theorem III.5.21].

Proposition 4.4 Let $f\colon X\to Y$ satisfy assumption (d) and $Y$ affine. Then categories ${\mathcal {D}}^b(X)$ and ${\mathcal {D}}^b(\mathop {{}^{p}{\rm Per}}\nolimits (X/Y))$ are equivalent, for $p=-1$ or $0$.

Proof. As objects in $\mathscr {P}_{p}$ are coherent sheaves on $X$, we have a functor $\Theta \colon {\rm Hot}^{-,b}(\mathscr {P}_{p})\to {\mathcal {D}}^b(X)$. There exists $N_0$ such that $\mathop {{\rm Ext}}\nolimits ^q_{X}(P, C) = 0$, for any $P \in \mathscr {P}_{p}$ and $C \in \mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$ and $q\geqslant N_0$. Indeed, $C$ has non-zero cohomology sheaves in degrees $-1$ and $0$ only, and objects in $\mathscr {P}_p$ are locally free sheaves on $X$ (see [VdB04, Proposition 3.2.6]). Then Proposition E.4 implies that cohomology groups of complex $\prod _{j-i = p} \mathop {{\rm Hom}}\nolimits _X(A^i, B^j)$ are isomorphic to $\mathop {{\rm Hom}}\nolimits _X(A^{\boldsymbol {\cdot }}, B^{\boldsymbol {\cdot }})$, for any $A^{\boldsymbol {\cdot }}, B^{\boldsymbol {\cdot }} \in \mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_p)$ (see Lemma 4.3). Hence, $\mathop {{\rm Hom}}\nolimits ^{\boldsymbol {\cdot }}_X(A^{\boldsymbol {\cdot }}, B^{\boldsymbol {\cdot }}) \simeq \mathop {{\rm Hom}}\nolimits _{\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_p)}(A^{\boldsymbol {\cdot }}, B^{\boldsymbol {\cdot }})$, i.e. $\Theta$ is fully faithful. It is essentially surjective, because $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$ is the heart of a bounded $t$-structure.

By Proposition 3.12, functor

(45)\begin{equation} \Sigma_o := (f^{+*}f_*(-))^{\vee \vee} \colon \mathscr{P}_{-1} \to \mathscr{P}_0^+ \end{equation}

is fully faithful. Denote by

(46)\begin{equation} \Sigma \colon {\rm Hot}^{-,b}(\mathscr{P}_{-1}) \to {\rm Hot}^{-}(\mathscr{P}_{0}^+) \end{equation}

the functor obtained by applying $\Sigma _o$ term-wise to complexes of objects of $\mathscr {P}_{-1}$.

Theorem 4.5 Let $f\colon X\to Y$ satisfy assumption (a) and $f^+ \colon X^+ \to Y$ be the flop of $f$. Then functor $\Sigma$ induces an equivalence between ${\mathcal {D}}^b(X)$ and ${\mathcal {D}}^b(X^+)$.

Proof. Lemma 3.11 implies that functor

\[ T_o := (f^*f^+_*(-))^{\vee \vee} \colon \mathscr{P}^+_0 \to \mathscr{P}_{-1} \]

is the inverse of $\Sigma _o$. Let us extend it term-wise to a functor

\[ T \colon {\rm Hot}^{-,b}(\mathscr{P}_0^+) \to {\rm Hot}^{-}(\mathscr{P}_{-1}). \]

Proposition 3.12 implies that $\Sigma$ and $T$ are inverse equivalences between $\mathscr {P}_{-1}$ and $\mathscr {P}_0^+$. Hence, they are also inverse equivalences between $\mathop {{\rm Hot}}\nolimits ^-(\mathscr {P}_{-1})$ and $\mathop {{\rm Hot}}\nolimits ^-(\mathscr {P}_0)$. By Proposition 4.4, we have ${\mathcal {D}}^b(X) \simeq {\rm Hot}^{-,b}(\mathscr {P}_{-1})$ and ${\mathcal {D}}^b(X^+) \simeq {\rm Hot}^{-,b}(\mathscr {P}_0^+)$. Thus, it suffices to show that $\Sigma$ takes $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})$ to $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_0^+)$ and $T$ takes $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{0}^+)$ to $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})$.

Let $E$ be in $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})$ and assume that $\Sigma (E)$ has unbounded cohomology. If cohomology of $\Sigma (E)$ is unbounded with respect to the standard $t$-structure it is also unbounded with respect to the $t$-structure with heart $\mathop {{}^{0}{\rm Per}}\nolimits (X^+/Y)$, because these two $t$-structures are related by a tilt. Let ${\mathcal {N}}^+$ be the projective generator of $\mathop {{}^{0}{\rm Per}}\nolimits (X^+/Y)$. Then $\dim _k \mathop {{\rm Hom}}\nolimits _{X^+}^{\boldsymbol {\cdot }}({\mathcal {N}}^+, \Sigma (E))$ is infinite. By adjunction of inverse functors, we have that $\dim _k \mathop {{\rm Hom}}\nolimits _X^{\boldsymbol {\cdot }}(T({\mathcal {N}}^+), E)$ is infinite. As $T_o({\mathcal {N}}^+)$ is in $\mathscr {P}_{-1}$, it contradicts the boundedness of the cohomology of $E$ with respect to the $t$-structure with heart $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$. Analogously, $T$ takes $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{0}^+)$ to $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})$.

Let us show that Theorem 4.5 may be considered as a write-up of Van den Bergh's theorem.

Let $f\colon X\to Y$ satisfy assumption (a). As $f$ has fibers of dimension bounded by one, [VdB04, Lemma 3.2.2] implies that for an $f$-ample line bundle ${\mathcal {L}}$ both ${\mathcal {L}} \oplus {\mathcal {O}}_X$ and ${\mathcal {O}}_X \oplus {\mathcal {L}}^{-1}$ are compact generators of ${\mathcal {D}}_{{\rm qc}}(X)$. It follows that both the projective generator ${\mathcal {M}} = \bigoplus {\mathcal {M}}_i$ of $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ (defined as in Remark 2.7) and ${\mathcal {N}}= \mathop {\mathcal {H}{\rm om}}\nolimits ({\mathcal {M}}, {\mathcal {O}}_X)$ of $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$ (see Remark 2.6) are compact generators of the category ${\mathcal {D}}_{{\rm qc}}(X)$. Analogous result holds for projective generators for $\mathop {{}^{-1}{\rm Per}}\nolimits (X^+/Y)$ and $\mathop {{}^{0}{\rm Per}}\nolimits (X^+/Y)$.

As $(f^{+*}f_*(-))^{\vee \vee }$ restricts to an equivalence $\mathscr {P}_{-1} \to \mathscr {P}_0^+$, Proposition 3.12, ${\mathcal {N}}^+$ (as in (35)) is a projective generator for $\mathop {{}^{0}{\rm Per}}\nolimits (X^+/Y)$ and the endomorphisms algebras of ${\mathcal {M}}$ and ${\mathcal {N}}^+$ are isomorphic.

Moreover, by Lemma 4.3, both ${\mathcal {M}}$ and ${\mathcal {N}}^+$ have no higher self Ext groups. Hence, by the result of Keller [Kel06] both ${\mathcal {D}}_{{\rm qc}}(X)$ and ${\mathcal {D}}_{{\rm qc}}(X^+)$ are equivalent to the derived category of DG modules over the algebra $A = \mathop {{\rm Hom}}\nolimits _X({\mathcal {M}},{\mathcal {M}})$, cf. [BVdB03]. Denote by

\[ \Sigma_{{\rm qc}} \colon {\mathcal{D}}_{{\rm qc}}(X) \xrightarrow{\simeq} {\mathcal{D}}_{{\rm qc}}(X^+) \]

the resulting equivalence.

Proposition 4.6 Let $f\colon X\to Y$ satisfy assumption (a). Functor $\Sigma _{{\rm qc}}|_{{\mathcal {D}}^b(X)}:{\mathcal {D}}^b(X)\to {\mathcal {D}}^b(X^+)$ is an equivalence that takes $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ to $\mathop {{}^{0}{\rm Per}}\nolimits (X^+/Y)$, and it coincides with Van den Bergh's equivalence in [VdB04, Theorem 4.4.2].

Proof. Category $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})$ is a subcategory of the homotopy category of complexes of projective $A$-modules; it consists of bounded above complexes. As $\Sigma _o$ as in (45) maps projective generator for $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ to the projective generator for $\mathop {{}^{0}{\rm Per}}\nolimits (X^+/Y)$, we have $\Sigma _{{\rm qc}}|_{\mathscr {P}_{-1}} = \Sigma _o$. We conclude that $\Sigma = \Sigma _{{\rm qc}}|_{\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})}$ is the equivalence defined by Van den Bergh in [VdB04, Theorem 4.4.2].

Categories $\mathscr {P}_{0}$ and $\mathscr {P}_{-1}^+$ are also equivalent. We can thus analogously construct an equivalence $\Sigma _{{\rm qc}}^{-1} \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X^+)$ that takes $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$ to $\mathop {{}^{-1}{\rm Per}}\nolimits (X^+/Y)$.

4.2 An alternative description of the flop functor

Consider diagram (42). The counit of $f^* \dashv f_*$ adjunction

\[ f^*f_* \to \mathop{{\rm Id}}\nolimits \]

gives a morphism

\[ p^*f^*f_* \to p^*. \]

Isomorphism $p^*f^* \simeq p^{+*}f^{+*}$ and $p^{+*}\dashv p^+_*$ adjunction lead to a base-change morphism

(47)\begin{equation} \epsilon \colon f^{+*}f_* \to p^+_*p^*. \end{equation}

Note that the morphism $\epsilon$ is, by definition, the composition

(48)\begin{equation} \epsilon \colon f^{+*}f_* \rightarrow p^+_*p^{+*}f^{+*}f_* \rightarrow p^{+}_*p^*. \end{equation}

Analogous argument gives a derived base change

(49)\begin{equation} \widetilde{\epsilon} \colon Lf^{+*}Rf_* \to Rp^+_* Lp^*. \end{equation}

Lemma 4.7 Let $f$ satisfy assumption (a) and ${\mathcal {M}}$ be an object in $\mathscr {P}_{-1}$. Then $\epsilon _{{\mathcal {M}}} \colon f^{+*}f_* {\mathcal {M}} \to p^+_*p^* {\mathcal {M}}$ is an isomorphism.

Proof. As ${\mathcal {M}}$ is locally free [VdB04, Propostion 3.2.6], applying $p^*$ to sequence (15) we obtain an exact sequence

\[ 0 \to p^* \mathcal{P} \to p^*f^*f_* {\mathcal{M}} \to p^*{\mathcal{M}} \to 0 \]

on $X\times _Y X^+$. Sheaf ${\mathcal {P}}$ is in $\mathscr {A}_f$, hence Proposition 2.11 implies an isomorphism

\[ p^+_*p^*f^*f_*{\mathcal{M}} \xrightarrow{\simeq} p^+_*p^* {\mathcal{M}}. \]

As diagram (42) is commutative, $p^+_*p^*f^*f_* {\mathcal {M}} \simeq p^+_*p^{+*}f^{+*}f_* {\mathcal {M}}$. Isomorphism $f^{+*}f_*{\mathcal {M}} \simeq f^{+*}f^+_* {\mathcal {N}}^+$ (see Proposition 3.12) together with Lemma 3.5 implies that $f^{+*}f_*{\mathcal {M}}$ is torsion-free. Moreover, by Remark 4.2, $Rp^+_* {\mathcal {O}}_{X\times _Y X^+} = {\mathcal {O}}_{X^+}$. Thus, conditions of Lemma 2.8 are satisfied and we have an isomorphism $f^{+*}f_* {\mathcal {M}} \xrightarrow {\simeq } p^+_*p^{+*}f^{+*}f_* {\mathcal {M}}$. Hence, $\epsilon _{{\mathcal {M}}}$ is a composite of two isomorphisms in (48).

Proposition 4.8 Let $f$ satisfy assumption (a). Then the flop functor $Rp^+_* Lp^*$ on $\mathscr {P}_{-1}$ is isomorphic to the non-derived flop functor $p^+_*p^*$.

Proof. Every object in $\mathscr {P}_{-1}$ locally free, [VdB04, Propostion 3.2.6], hence $Lp^*|_{ \mathscr {P}_{-1}} \simeq p^*|_{\mathscr {P}_{-1}}$. Moreover, because $R^1f_* \mathscr {P}_{-1} \simeq 0$, [VdB04, Lemma 3.1.2] Proposition 2.11 implies that $R^1p^+_*p^* \mathscr {P}_{-1} \simeq 0$.

We define functor

\[ f^{+*}f_* \colon {\rm Hot}^{-,b}(\mathscr{P}_{-1}) \to {\rm Hot}^{-}(f^{+*}f_* \mathscr{P}_{-1}) \]

term-wise. We use the same notation for the composite of the above functor with the canonical functor ${\rm Hot}^{-}(f^{+*}f_* \mathscr {P}_{-1}) \to {\mathcal {D}}^-(X^+)$. Also term-wise, we define a functor

\[ p^+_*p^* \colon {\rm Hot}^{-,b}(\mathscr{P}_{-1}) \to {\mathcal{D}}^-(X^+). \]

Lemma 4.9 Let $f$ satisfy assumption (a). Then the functor $f^{+*}f_*$ maps ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$ to ${\mathcal {D}}^b(X^+)$. The functor $f^*f_*$ maps ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$ to ${\mathcal {D}}^b(X)$.

Proof. The $i^*\dashv i_*$ adjunction counit induces an isomorphism $g^{+*}g_*\xrightarrow {\simeq }f^{+*}f_*$. Hence, it is enough to show that the term-wise extension of $g^{+*}g_*$ maps ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$ to ${\mathcal {D}}^b(X^+)$.

Functor $Rg_*$ preserves the bounded derived category of coherent sheaves, because $g$ is proper, and so does $Lg^{+*}$, because ${\mathcal {Y}}$ is smooth. Hence, for $P^{\boldsymbol {\cdot }}$ in ${\rm Hot}^{-,b}(\mathscr {P}_{-1}) \simeq {\mathcal {D}}^b(X)$ (see Proposition 4.4), $Lg^{+*}Rg_*(P^{\boldsymbol {\cdot }})$ is an object in ${\mathcal {D}}^b(X^+)$. For $P \in \mathscr {P}_{-1}$, we denote by $P^+ = (f^{+*}f_*P)^{\vee \vee }$ the corresponding object in $\mathscr {P}^+_0$ (see Proposition 3.12). As $f^+_*P^+ = f_*P$ and $g^+_*P^+ = g_*P$, Proposition 3.15 for $g^+$, $f^+$ and $P^+$ instead of $g$, $f$ and ${\mathcal {R}}$, implies that the first sheet of the spectral sequence $E^{p,q}_1 = {\mathcal {H}}^q_{X^+}(Lg^{+*}g_* P^p) \Rightarrow {\mathcal {H}}^{p+q}_{X^+}(Lg^{+*}Rg_* P^{{\boldsymbol {\cdot }}})$ has two non-zero rows:

\[ E_1^{\bullet,0} = f^{+*}f^+_*P^{+\bullet} = g^{+*}g^+_*P^{+\bullet} =g^{+*}g_*P^{\bullet},\quad E_1^{\bullet, -1}= P^{+\bullet}. \]

By Theorem 4.5, the complex $P^{+\bullet }$ is in ${\rm Hot}^{-,b}(\mathscr {P}_0^+)$, hence cohomology of the -1st row is bounded: $E_2^{N,-1} = 0$, for $N\leqslant N_0$. The spectral sequence implies that, for sufficiently negative $N$, ${\mathcal {H}}_{X^+}^{N}(g^{+*}g_*P^\bullet ) = E_2^{N,0} = E_{\infty }^{N,0} = {\mathcal {H}}_{X^+}^{N}(Lg^{+*}g_*P^\bullet )=0$, i.e. $g^{+*}g_*(P^\bullet ) \in {\mathcal {D}}^b(X^+)$.

The proof of the boundedness for $f^*f_*(P^\bullet ) = g^*g_*(P^\bullet )$ is similar with the use of Proposition 3.15 for ${\mathcal {M}}$ instead of ${\mathcal {N}}^+$.

Proposition 4.10 Let $f$ satisfy assumption (a). Then functors $f^{+*} f_*$ and $p^+_*p^*$ are isomorphic on the category ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$. They are also isomorphic to the flop functor restricted to ${\mathcal {D}}^b(X)$ under the equivalence ${\mathcal {D}}^b(X) \simeq {\rm Hot}^{-,b}(\mathscr {P}_{-1})$.

Proof. Proposition 4.8 implies that the canonical morphism $p^+_*p^*|_{\mathscr {P}_{-1}} \to Rp^+_* p^*|_{\mathscr {P}_{-1}}$ is an isomorphism. All objects in $\mathscr {P}_{-1}$ are locally free [VdB04, Propostion 3.2.6], hence canonical morphism $Rp^+_*Lp^*|_{\mathscr {P}_{-1}} \to Rp^+_*p^*|_{\mathscr {P}_{-1}}$ is also an isomorphism. This implies isomorphism of $p^+_*p^*|_{\mathscr {P}_{-1}}$ and $Rp^+_*Lp^*|_{\mathscr {P}_{-1}}$.

Lemma 4.7 assures that the base change $\epsilon _{{\mathcal {M}}} \colon f^{+*}f_* {\mathcal {M}} \to p^+_*p^* {\mathcal {M}}$ is an isomorphism, for all ${\mathcal {M}}\in \mathscr {P}_{-1}$.

Thus, all three functors are isomorphic on $\mathscr {P}_{-1}$. Induction on triangles shows that the isomorphism extends to $\mathop {{\rm Hot}}\nolimits ^b(\mathscr {P}_{-1})$. By Lemmas 4.9 and 4.11, they are also isomorphic on ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$.

In the proof of Proposition 4.10 we use the fact that in order to check that a natural transformation gives an isomorphism of two functors defined on ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$, it suffices to check that it gives an isomorphism of the functors restricted to $\mathop {{\rm Hot}}\nolimits ^b(\mathscr {P}_{-1})$, which we prove now.

Lemma 4.11 Let ${\mathcal {A}}$, ${\mathcal {B}}$ be abelian categories, $\mathscr {P}\subset {\mathcal {B}}$ a full subcategory closed under extensions and $F, G \colon {\rm Hot}_{{\mathcal {B}}}^{-,b}(\mathscr {P}) \to {\mathcal {D}}^-({\mathcal {A}})$ exact functors. Assume there exists $n_0$ such that both $F$ and $G$ map ${\rm Hot}_{{\mathcal {B}}}^{-,b}(\mathscr {P})^{\leqslant 0}$ to ${\mathcal {D}}^-({\mathcal {A}})^{\leqslant n_0}$ and that the image of either $F$ or $G$ is contained in ${\mathcal {D}}^b({\mathcal {A}})$. Let $\kappa \colon F|_{{\rm Hot}_{{\mathcal {B}}}^b(\mathscr {P})} \to G|_{{\rm Hot}_{{\mathcal {B}}}^b(\mathscr {P})}$ be an isomorphism of functors. Then $\kappa$ admits an extension to a functorial isomorphism $\widetilde {\kappa } \colon F\to G$.

Proof. Let $E$ be an object in ${\rm Hot}^{-,b}(\mathscr {P})$. We denote by $\sigma _{\geqslant k}E$ the ‘stupid’ truncation:

\[ \sigma_{\geqslant k}(E)^i = \begin{cases}E^i, & \text{if }\, i \geqslant k,\\ 0, & {\rm otherwise.} \end{cases} \]

Morphism $\sigma _{\geqslant l}E \to E$ induces isomorphisms $\tau _{\geqslant m} F(\sigma _{\geqslant l}E) \xrightarrow {\simeq } \tau _{\geqslant m} F(E)$, $\tau _{\geqslant m} G(\sigma _{\geqslant l}E) \xrightarrow {\simeq } \tau _{\geqslant m} G(E)$, for any $m\in \mathbb {Z}$, and $l\leqslant m-n_0$. We have

(50)\begin{equation} \tau_{\geqslant m} F(E) \simeq \tau_{\geqslant m} F(\sigma_{\geqslant l} E) \simeq \tau_{\geqslant m} G(\sigma_{\geqslant l} E) \simeq \tau_{\geqslant m} G(E), \end{equation}

for any $l\leqslant m-n_0$. Let us assume that $F(E)$ is an object in ${\mathcal {D}}^b({\mathcal {A}})$. There exists $N$ such that morphism $F(E) \to \tau _{\geqslant m} F(E)$ is an isomorphism, for any $m\leqslant N$. Thus, isomorphisms (50) imply that $G(E)$ is also an object in ${\mathcal {D}}^b({\mathcal {A}})$. We define $\widetilde {\kappa }$ as the composite of isomorphisms

\[ F(E) \rightarrow \tau_{\geqslant m} F(E) \rightarrow \tau_{\geqslant m} F(\sigma_{\geqslant l} E) \xrightarrow{\tau_{\geqslant m}\kappa_{\sigma_{\geqslant l}E} }\tau_{\geqslant m} G(\sigma_{\geqslant l} E) \rightarrow \tau_{\geqslant m} G(E) \rightarrow G(E), \]

for any $m\leqslant N$ and $l\leqslant m-n_0$. As $\kappa$ is a natural transformation, $\tau _{\geqslant m} \kappa _{\sigma _{\geqslant l}E} \simeq \tau _{\geqslant m} \kappa _{\sigma _{\geqslant l'} E}$, for any $l,l'\leqslant m-n_0$. Hence, $\widetilde {\kappa }$ does not depend on the choice of $m\leqslant N$ and $l\leqslant m-n_0$.

4.3 A functorial exact triangle of functors ${\mathcal {D}}_{{\rm qc}}(X) \to {\mathcal {D}}_{{\rm qc}}(X^+)$

Assume that $f\colon X\to Y$ satisfies assumption (a), i.e. there exists a closed embedding $i\colon Y \to {\mathcal {Y}}$, for a smooth ${\mathcal {Y}}$ of dimension $n+1$. Then, $X\times _Y X^+\simeq X\times _{{\mathcal {Y}}} X^+$, i.e. diagram

(51)

is fibered.

Recall, that we denote by $F= Rp^+_*Lp^*$ the flop functor and by $\Sigma \colon \mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})\to {\mathcal {D}}^b(X^+)$ the term-wise extension of $\Sigma _o = (f^{+*}f_*(-))^{\vee \vee }\colon \mathscr {P}_{-1} \to \mathscr {P}_0^+$.

Lemma 4.12 Let $f\colon X\to Y$ satisfy assumption (a). Functors $Rf^+_* \Sigma,Rf_*\colon \mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1}) \to {\mathcal {D}}^b(Y)$ are isomorphic.

Proof. Consider morphism $\alpha \colon f_*|_{\mathscr {P}_{-1}}\to f^+_* \Sigma _o|_{\mathscr {P}_{-1}}$ defined as the composite

\[ f_* \xrightarrow{\eta f_*} f^+_*f^{+*}f_* \xrightarrow{f^+_* \beta f^{+*}f_*} f^+_*(f^{+*}f_*(-)^{\vee \vee}) = f^+_* \Sigma_o, \]

for the unit $\eta$ of $f^{+*}\dashv f^+_*$ adjunction and the reflexification $\beta \colon (-) \to (-)^{\vee \vee }$. Lemma 3.11 implies that $\alpha$ is an isomorphism of functors (see Lemma 4.11).

Morphism $\alpha$ yields morphism $\alpha ^{-} \colon f_*^-\to (f^{+}_* \Sigma )$ of the term-wise extension of functors $f_*|_{\mathscr {P}_{-1}}$ and $f^{+}_* \Sigma _o|_{\mathscr {P}_{-1}}$ to the category $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})$. As $\alpha$ is an isomorphism, the same is true about $\alpha ^-$.

Now let $E$ be an object in ${\mathcal {D}}^b(X)$. It is isomorphic to a complex ${\mathcal {P}}_{{\boldsymbol {\cdot }}}$ in $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})$. The first layer of spectral sequence

\[ E^1_{p,q} = R^pf_* {\mathcal{P}}_q \Rightarrow R^{p+q}f_* E \]

has only one non-zero row. Hence, complex $f_* {\mathcal {P}}_{{\boldsymbol {\cdot }}}$ is isomorphic to $Rf_* E$. Analogously, because $\Sigma _o(\mathscr {P}_{-1}) \simeq \mathscr {P}_0^+$, complex $f^+_* \Sigma ({\mathcal {P}}_{\boldsymbol {\cdot }})$ is isomorphic to $Rf^+_* \Sigma \, E$. Thus, $\alpha ^-$ induces an isomorphism of $Rf_*$ and $Rf^+_* \Sigma$.

Proposition 4.13 Let $f\colon X\to Y$ satisfy assumption (a) and let $\Sigma _{{\rm qc}}$ be as in Proposition 4.6. The derived base change $\widetilde {\epsilon }$ induces a functorial exact triangle of functors ${\mathcal {D}}_{{\rm qc}}(X) \to {\mathcal {D}}_{{\rm qc}}(X^+)$:

\[ \Sigma_{{\rm qc}}[1] \to Lg^{+*}Rg_* \xrightarrow{\widetilde{\epsilon}} F \to \Sigma_{{\rm qc}}[2]. \]

Proof. We have already seen in § 4.1 that there exists a choice of compact generators ${\mathcal {M}}$ and ${\mathcal {N}}^+$ of ${\mathcal {D}}_{{\rm qc}}(X)$ and ${\mathcal {D}}_{{\rm qc}}(X^+)$, respectively, such that ${\mathcal {D}}_{{\rm qc}}(X) \simeq {\mathcal {D}}(\text{Mod--}A) \simeq {\mathcal {D}}_{{\rm qc}}(X^+)$, for an algebra $A =\mathop {{\rm Hom}}\nolimits _X({\mathcal {M}}, {\mathcal {M}})$ (see Lemma 4.3). Under these equivalences, functor $\Sigma _{{\rm qc}}$ is given by algebra $A$ considered as an $A^{\mathop {{\rm op}}\nolimits }\otimes A$ bimodule.

Appendix C, allows us to lift the base change $Lg^{+*}Rg_* \to Rp^+_* Lp^*$ to a 1-morphism in $\mathop {\textbf {Bimod}}\nolimits$. It induces a functorial exact triangle

(52)\begin{equation} \Sigma' \to Lg^{+*}Rg_* \to Rp^+_* Lp^* \to \Sigma'[1]. \end{equation}

Let us show that functors $\Sigma '|_{\mathscr {P}_{-1}}$ and $\Sigma _o[1]|_{\mathscr {P}_{-1}}$ are isomorphic. To this end, we consider an object ${\mathcal {M}}$ in $\mathscr {P}_{-1}$. By Proposition 3.12, object ${\mathcal {N}}^+ := (f^{+*}f_*({\mathcal {M}}))^{\vee \vee }$ lies in $\mathscr {P}_0^+$. Thus, Proposition 3.15 yields an exact triangle

(53)\begin{equation} {\mathcal{N}}^+[1] \to Lg^{+*}Rg_*^+ {\mathcal{N}}^+ \to f^{+*}f^+_* {\mathcal{N}}^+ \to {\mathcal{N}}^+[2]. \end{equation}

By Lemma 3.11, we have $f^+_*{\mathcal {N}}^+\simeq f_* {\mathcal {M}}$, hence $g^+_* {\mathcal {N}}^+ \simeq g_* {\mathcal {M}}$. Proposition 4.10 assures that $f^{+*}f_*{\mathcal {M}} \simeq F({\mathcal {M}})$, hence triangle (53) reads

(54)\begin{equation} \Sigma_o({\mathcal{M}})[1] \to Lg^{+*}Rg_* {\mathcal{M}} \to F({\mathcal{M}}) \to \Sigma_o({\mathcal{M}})[2]. \end{equation}

There exists a morphism from triangle (54) to triangle (52) applied to ${\mathcal {M}}$ which is equal to the identity morphism on $Lg^{+*}g_*\, {\mathcal {M}}$ and on $F({\mathcal {M}}) = Rp^+_*Lp^*{\mathcal {M}}$. Indeed, the map $Lg^{+*}Rg_*{\mathcal {M}} \to F({\mathcal {M}})$ in (54) is the composite

\[ Lg^{+*}Rg_*{\mathcal{M}} \xrightarrow{\tau_{\geqslant 0}} f^{+*}f_*{\mathcal{M}} \xrightarrow{\simeq} p^+_*p^*{\mathcal{M}} \xrightarrow{\simeq} Rp^+_*Lp^*{\mathcal{M}} \]

of the truncation $\tau _{\geqslant 0}$, which in this case is taking the zeroth cohomology, followed by the base-change isomorphism (see Proposition 4.10) and then by the isomorphism $p^+_*p^*{\mathcal {M}} \simeq Rp^+_*Lp^*{\mathcal {M}}$ (see Proposition 4.8). Note that, under the isomorphism $f^{+*}f_* \simeq g^{+*}g_*$ of functors $\mathop {{\rm Coh}}\nolimits (X) \to \mathop {{\rm Coh}}\nolimits (X^+)$, induced by the $i^*\dashv i_*$ adjunction counit, the base change $f^{+*}f_* \to p^+_*p^*$ corresponds to the base change $g^{+*}g_* \to p^+_*p^*$. On the other hand, the map $Lg^{+*}Rg_* \to Rp^+_*Lp^*$ in (52) is the (derived) base change. For the degree reason (as $Rp^+_*Lp^*{\mathcal {M}}$ is a sheaf by Proposition 4.8), the derived base change for ${\mathcal {M}}$ factors via $H^0(Lg^{+*}Rg_*{\mathcal {M}})$. The existence of the desired morphism of triangles follows from the fact that the restriction of the derived base change to zeroth cohomology is the non-derived base change. Thus, for any ${\mathcal {M}}$ in $\mathscr {P}_{-1}$, there exists an isomorphism $\alpha _{{\mathcal {M}}} \colon \Sigma _o({\mathcal {M}})[1] \to \Sigma '({\mathcal {M}})$. It is unique, as $\mathop {{\rm Hom}}\nolimits _{X^+}(\Sigma _o({\mathcal {M}})[1], F({\mathcal {M}})[-1]) = \mathop {{\rm Ext}}\nolimits ^{-2}_{X^+}({\mathcal {N}}^+, f^{+*}f^+_* {\mathcal {N}}^+) = 0$ (for degree reasons). Morphism $\alpha _{{\mathcal {M}}}$ when composed with $\Sigma '({\mathcal {M}}) \to Lg^{+*}Rg_*({\mathcal {M}})$ is equal to $\beta _{{\mathcal {M}}}$, for a morphism $\beta = \eta \Sigma _o \colon \Sigma _o[1] \to Lg^{+*} Rg_*$. Here, $\eta \colon \mathop {{\rm Id}}\nolimits [1] \to g^{+!}Rg^+_*[1] \simeq Lg^{+*}Rg^+_*$ is the adjunction unit (we use $Rg^+_*\Sigma _o \simeq Rg_*$ which follows from Lemma 4.12).

For any ${\mathcal {M}}, {\mathcal {M}}_1 \in \mathscr {P}_{-1}$ and $\varphi \colon {\mathcal {M}} \mapsto {\mathcal {M}}_1$, both compositions

\[ \Sigma_o({\mathcal{M}})[1] \xrightarrow{\Sigma_o(\varphi)[1]} \Sigma_o({\mathcal{M}}_1)[1] \xrightarrow{\alpha_{{\mathcal{M}}_1}} \Sigma'({\mathcal{M}}_1),\quad \Sigma_o({\mathcal{M}})[1] \xrightarrow{\alpha_{{\mathcal{M}}}} \Sigma'({\mathcal{M}}) \xrightarrow{\Sigma'(\varphi)} \Sigma'({\mathcal{M}}_1) \]

fit into the following commuting diagram.

As both $\mathop {{\rm Hom}}\nolimits _{X^+}(\Sigma _o({\mathcal {M}})[1], F({\mathcal {M}}_1))$ and $\mathop {{\rm Hom}}\nolimits _{X^+}(\Sigma _o({\mathcal {M}})[1], F({\mathcal {M}}_1)[-1])$ vanish (also for degree reasons), we have $\Sigma '(\varphi ) \circ \alpha _{{\mathcal {M}}} = \alpha _{{\mathcal {M}}_1} \circ \Sigma _o(\varphi )[1]$, i.e. $\alpha$ extends to a morphism of functors $\alpha \colon \Sigma _o[1]|_{\mathscr {P}_{-1}} \to \Sigma '|_{\mathscr {P}_{-1}}$. Moreover, any $\psi$ in $\mathop {{\rm Hom}}\nolimits _{X^+}(\Sigma _o({\mathcal {M}})[1], \Sigma _o({\mathcal {M}}_1)[1]) \simeq \mathop {{\rm Hom}}\nolimits _X({\mathcal {M}}, {\mathcal {M}}_1)$ uniquely determines $\widetilde {\psi } \colon \Sigma '({\mathcal {M}}) \to \Sigma '({\mathcal {M}}_1)$. Similarly, as $\mathop {{\rm Hom}}\nolimits _{X^+}(\Sigma '({\mathcal {M}}), F({\mathcal {M}}_1)) = 0 =\mathop {{\rm Hom}}\nolimits _{X^+}(\Sigma '({\mathcal {M}}), F({\mathcal {M}}_1)[-1])$, any $\tau \in \mathop {{\rm Hom}}\nolimits _{X^+}(\Sigma '({\mathcal {M}}), \Sigma '({\mathcal {M}}_1))$ determines $\widetilde {\tau } \colon \Sigma _o({\mathcal {M}})[1] \to \Sigma _o({\mathcal {M}}_1)[1]$. Thus, morphism $\alpha \colon \mathop {{\rm Hom}}\nolimits _{X^+}(\Sigma _o({\mathcal {M}})[1], \Sigma _o({\mathcal {M}}_1)[1])\to \mathop {{\rm Hom}}\nolimits _{X^+}(\Sigma '({\mathcal {M}}), \Sigma '({\mathcal {M}}_1))$ is a bijection. It follows that $\alpha$ is an isomorphism of functors $\alpha \colon \Sigma _o[1]|_{\mathscr {P}_{-1}} \xrightarrow {\simeq } \Sigma '|_{\mathscr {P}_{-1}}$.

Lemma 4.3 implies that ${\mathcal {D}}_{qc}(X) \simeq {\mathcal {D}}(\text{Mod--}A)$, for an algebra $A$. Furthermore, by definition, functor $\Sigma _{{\rm qc}}$ is isomorphic to the bimodule functor $\Phi _A$, for $A$ considered as $A^{\mathop {{\rm op}}\nolimits }\otimes A$ bimodule. By construction (via an exact triangle), $\Sigma '$ is also a bimodule functor. Thus, assumptions of Lemma C.12 are satisfied and we conclude that $\Sigma '[-1] \simeq \Sigma _{{\rm qc}}$. Hence, functorial exact triangle (52) reads

\[ \Sigma_{{\rm qc}}[1] \to Lg^{+*}Rg_* \to F \to \Sigma_{{\rm qc}}[2]. \]

Corollary 4.14 Let $f\colon X\to Y$ satisfy assumption (p). The flop functor $F$ takes ${\mathcal {D}}^b(X)$ to ${\mathcal {D}}^b(X^+)$.

Proof. The statement is local in $Y$, therefore we can assume that morphism $f$ satisfies assumption (a). As both $Lg^{+*}Rg_*$ and $\Sigma _{{\rm qc}}$ take ${\mathcal {D}}^b(X)$ to ${\mathcal {D}}^b(X^+)$ (see Theorem 4.5), the functorial exact triangle of Proposition 4.13 implies that the flop functor $F$ takes ${\mathcal {D}}^b(X)$ to ${\mathcal {D}}^b(X^+)$.

4.4 An alternative description of the flop–flop functor

We show that $F^+F$ on category ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$ is isomorphic to $f^*f_*$. First, we show vanishing of higher inverse images of projective objects in $\mathscr {A}_f$ and then extend Proposition 4.8 to the sheaf $f^*f_* {\mathcal {N}}$.

Lemma 4.15 Let $f\colon X\to Y$ satisfy assumption (a) and ${\mathcal {P}}$ be projective in $\mathscr {A}_f$. Then $L^jp^* {\mathcal {P}} = 0$, for $j\geqslant 2$.

Proof. The statement is local in $X$, in particular in $Y$. Therefore, we can assume that $f$ satisfies assumption (c) and ${\mathcal {P}}$ is a direct sum of copies of ${\mathcal {P}}_i$ as in (19).

Consider the counit $Lg^*g_* {\mathcal {M}}_i \to {\mathcal {M}}_i$ of the $Lg^* \dashv Rg_*$ adjunction. It gives an exact triangle

(55)\begin{equation} Lg^*g_* {\mathcal{M}}_i \to {\mathcal{M}}_i \to S_{{\mathcal{M}}_i} \to Lg^*g_* {\mathcal{M}}_i[1]. \end{equation}

Applying functor $Lp^*$ to it yields an exact triangle on $X\times _Y X^+$:

\[ L(p^*g^*)g_*{\mathcal{M}}_i \to p^* {\mathcal{M}}_i \to Lp^* S_{{\mathcal{M}}_i} \to L(p^*g^*)g_* {\mathcal{M}}_i[1]. \]

Lemma 3.14 implies that $L^j(p^*g^*)g_* {\mathcal {M}}_i = 0$, for $j\geqslant 2$. Thus, $L^j p^* S_{{\mathcal {M}}_i} = 0$, for $j>2$.

Proposition 3.15, the long exact sequence of cohomology associated to triangle (55) and sequence (15) imply that $S_{{\mathcal {M}}_i}$ has two non-zero cohomology sheaves:

\[ {\mathcal{M}}_i[2] \to S_{{\mathcal{M}}_i} \to {\mathcal{P}}_i[1] \to {\mathcal{M}}_i[3]. \]

By applying $Lp^*$ and looking at the cohomology sheaves of the obtained triangle, we obtain an isomorphism $L^jp^*{\mathcal {P}}_i \simeq L^{j+1}p^* S_{{\mathcal {M}}_i} \simeq 0$, for $j\geqslant 3$, and an exact sequence

\[ 0 \to L^2p^* {\mathcal{P}}_i \to p^* {\mathcal{M}}_i \to L^2 p^* S_{{\mathcal{M}}_i} \to L^1p^* {\mathcal{P}}_i \to 0. \]

Sheaf $L^2p^* {\mathcal {P}}_i$ is supported on the exceptional divisor of $p$, hence it is torsion. As $p^* {\mathcal {M}}_i$ is locally free, $L^2 p^* {\mathcal {P}}_i = 0$.

Lemma 4.16 Let $f\colon X\to Y$ satisfy assumption (a). Then, for any ${\mathcal {N}} \in \mathscr {P}_0$, the pullbacks of ${\mathcal {N}}$ and ${\mathcal {M}}^+ = (f^{+*}f_*{\mathcal {N}})^{\vee \vee }$ fit into the short exact sequence

(56)\begin{equation} 0 \to p^{+*}{\mathcal{M}}^+ \to p^*{\mathcal{N}}\to L^1p^{+*}(f^{+*}f^+_{*}{\mathcal{M}}^+) \to 0 \end{equation}

on $X\times _Y X^+$.

Proof. By Proposition 3.15, we have an exact triangle

(57)\begin{equation} {\mathcal{N}}[1] \to Lg^*g_*{\mathcal{N}} \to f^*f_*{\mathcal{N}} \to {\mathcal{N}}[2]. \end{equation}

By applying $Lp^*$ to (57), we obtain an exact triangle

(58)\begin{equation} Lp^* {\mathcal{N}}[1] \to L(p^*g^*)g_* {\mathcal{N}}\to Lp^*(f^*f_* {\mathcal{N}}) \to Lp^*{\mathcal{N}}[2]. \end{equation}

As $g_* {\mathcal {N}}$ has a locally free resolution of length two (see Lemma 3.14), $L^2(p^*g^*)g_*{\mathcal {N}} =0$. The long exact sequence of cohomology for triangle (58) implies the exact sequence

(59)\begin{equation} 0 \to L^2p^*(f^*f_* {\mathcal{N}}) \to p^* {\mathcal{N}}\to L^1(p^*g^*)g_* {\mathcal{N}} \to L^1p^*(f^*f_* {\mathcal{N}}) \to 0. \end{equation}

As ${\mathcal {N}}$ is locally free, $L^{>0}p^*{\mathcal {N}} =0$. Hence, by applying $Lp^*$ to sequence (32), we obtain isomorphisms $L^3p^*\mathcal {Q} \simeq L^2p^*(f^*f_* {\mathcal {N}})$ and $L^2 p^* \mathcal {Q} \simeq L^1 p^*(f^*f_* {\mathcal {N}})$, where $\mathcal {Q} = {\mathcal {H}}_X^0(\iota _{f*}\iota _f^* {\mathcal {N}})$ is a projective object in $\mathscr {A}_f$. In view of Lemma 4.15, sequence (59) is reduced to an isomorphism $p^* {\mathcal {N}}\simeq L^1(p^*g^*)g_* {\mathcal {N}}$.

Let ${\mathcal {M}}^+ = (f^{+*}f_* {\mathcal {N}})^{\vee \vee }$ be an object in $\mathscr {P}_{-1}^+$. Proposition 3.12 implies that $f^+_* {\mathcal {M}}^+ \simeq f_* {\mathcal {N}}$, hence $g^+_* {\mathcal {M}}^+ \simeq g_* {\mathcal {N}}$. As diagram (51) commutes, we have

(60)\begin{equation} p^* {\mathcal{N}} \simeq L^1(p^*g^*)g_* {\mathcal{N}} \simeq L^1(p^{+*}g^{+*})g^+_* {\mathcal{M}}^+. \end{equation}

By applying $Lp^{+*}$ to sequence (36), we obtain an exact sequence

(61)\begin{equation} 0 \to L^2 p^{+*}(f^{+*}f^+_* {\mathcal{M}}^+) \to p^{+*}{\mathcal{M}}^+ \to L^1(p^{+*}g^{+*})g^+_* {\mathcal{M}}^+ \to L^1p^{+*}(f^{+*}f^+_*{\mathcal{M}}^+) \to 0. \end{equation}

Now consider sequence (15) with $f^+$, ${\mathcal {M}}^+$ and ${\mathcal {P}}^+$ instead of $f$, ${\mathcal {M}}$ and ${\mathcal {P}}$. Apply functor $Lp^{+*}$ to it. Then Lemma 4.15 for $f^+$, $p^+$ and ${\mathcal {P}}^+$ instead of $f$,$p$ and ${\mathcal {P}}$, implies that $L^2p^{+*}(f^{+*}f^+_* {\mathcal {M}}^+) \simeq 0$. Hence, sequence (61) can be rewritten as

(62)\begin{equation} 0 \to p^{+*}{\mathcal{M}}^+ \to L^1(p^{+*}g^{+*})g^+_* {\mathcal{M}}^+ \to L^1p^{+*}(f^{+*}f^+_*{\mathcal{M}}^+) \to 0. \end{equation}

Sequence (62) together with isomorphism (60) imply that (56) is exact.

Proposition 4.17 Let $f$ satisfy assumption (a). Then flop functor $Rp^+_* Lp^*$ on category $f^*f_* \mathscr {P}_0$ is isomorphic to the non-derived flop functor $p^+_*p^*$.

Proof. In view of Lemma 4.15, $L^jp^* \mathcal {Q} \simeq 0$, for any $\mathcal {Q}$ projective in $\mathscr {A}_f$ and any $j\geqslant 2$. As any object ${\mathcal {N}}$ in $\mathscr {P}_0$ is locally free [VdB04, Proposition 3.2.6], by applying $Lp^*$ to sequence (32), we obtain an isomorphism $Lp^* f^*f_*{\mathcal {N}} \simeq p^*f^*f_* {\mathcal {N}}$. As diagram (42) commutes, the latter sheaf is isomorphic to $p^{+*}f^{+*}f_* {\mathcal {N}}$. Lemma 2.8 implies that $R^1p^+_* p^{+*}(E) \simeq 0$, for any sheaf $E$ on $X^+$. Thus, $Rp^+_* p^{+*}f^{+*}f_* {\mathcal {N}}\simeq p^+_* p^{+*}f^{+*}f_* {\mathcal {N}} \simeq p^+_* p^{*}f^{*}f_* {\mathcal {N}}$.

Base change $\epsilon$ given by (47) and an analogous $\epsilon ^+ \colon f^*f^+_* \to p_* p^{+*}$ together with the adjunction unit $\eta ^{f^+}\colon \mathop {{\rm Id}}\nolimits \to f^+_*f^{+*}$ give $\widetilde {\epsilon } \colon f^*f_* \to p_*p^{+*}p^+_*p^*$ which is the composite

(63)\begin{equation} \widetilde{\epsilon} \colon f^*f_* \xrightarrow{f^*\eta^{f^+}f_*} f^*f^+_*f^{+*}f_* \xrightarrow{\epsilon^+(f^{+*}f_*)} p_*p^{+*}f^{+*}f_* \xrightarrow{(p_*p^{+*}) \epsilon} p_*p^{+*}p^+_*p^*. \end{equation}

Proposition 4.18 Let $f$ satisfy assumption (a). Morphism $\widetilde {\epsilon }$ is an isomorphism on the category ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$.

Proof. We check that, for any ${\mathcal {M}}\in \mathscr {P}_{-1}$, the map $\widetilde {\epsilon }_{{\mathcal {M}}}$ is a composition of three isomorphisms.

Take ${\mathcal {N}}^+:= (f^{+*}f_* {\mathcal {M}})^{\vee \vee }$ as in (35). In view of Lemma 3.11, $f_* {\mathcal {M}} \simeq f^+_* {\mathcal {N}}^+$. Then Lemma 2.9 implies that morphism $\eta ^{f^+}_{f_*{\mathcal {M}}}\colon f_* {\mathcal {M}} \to f^+_*f^{+*}f_* {\mathcal {M}}$ is an isomorphism, hence so is $f^* \eta ^{f^+}f_*$ on ${\mathcal {M}}$.

Natural transformation $\epsilon ^+$ is the composite of the adjunction unit $\eta ^p$ and the counit $\varepsilon ^{f^+}$ as in (48). We check that both maps

\begin{align*} &f^*f^+_*f^{+*}f_*{\mathcal{M}} \xrightarrow{\eta^p_{f^*f^+_*f^{+*}f_*{\mathcal{M}}}} p_*p^*f^*f^+_*f^{+*}f_*{\mathcal{M}}, \\ &p_*p^{+*}f^{+*}f^+_*f^{+*}f_*{\mathcal{M}} \xrightarrow{p_*p^{+*}\varepsilon^{f^+}_{f^{+*}f_*{\mathcal{M}}}} p_*p^{+*}f^{+*}f_*{\mathcal{M}} \end{align*}

are isomorphisms. As $f^*f^+_*f^{+*}f_* {\mathcal {M}} \simeq f^*f_* {\mathcal {M}}$, morphism $\eta ^p_{f^*f^+_*f^{+*}f_*{\mathcal {M}}}$ is an isomorphism if and only if $\eta ^p_{f^*f_* {\mathcal {M}}}$ is also an isomorphism. Object ${\mathcal {N}}^+$ satisfies $R^1f^+_*{\mathcal {N}}^+ = 0$, hence, by Proposition 2.11, $R^1p_*p^{+*}{\mathcal {N}}^+ =0$. As, by Lemma 2.8, $R^ip_*p^*{\mathcal {M}} =0$, for $i\geqslant 1$, applying $Rp_*$ to sequence (56) with the roles of $f$ and $f^+$ exchanged gives $R^1p_*L^1p^*(f^*f_*{\mathcal {M}}) \simeq R^1p_*p^{+*}{\mathcal {N}}^+ =0$. Then, short exact sequence (20), for $E=f^*f_*{\mathcal {M}}$ and $p$ instead of $f$, implies that the adjunction unit $\eta ^p_{f^*f_* {\mathcal {M}}}\colon f^*f_*{\mathcal {M}} \xrightarrow {\simeq } p_*p^*(f^*f_*{\mathcal {M}})$ is an isomorphism.

By the definition of the adjunction, morphism $\varepsilon ^{f^+}_{f^{+*}f_*{\mathcal {M}}}$ fits into the following commutative triangle.

We have already seen that $\eta ^{f^+}_{f_*{\mathcal {M}}}$ is an isomorphism, hence the same is true about $f^{+*}\eta ^{f^+}_{f_*{\mathcal {M}}}$. As, clearly, $\mathop {{\rm Id}}\nolimits _{f^{+*}f_*{\mathcal {M}}}$ is an isomorphism, so is the third morphism in the triangle $\varepsilon ^{f^+}_{f^{+*}f_*{\mathcal {M}}}$.

Finally, by Lemma 4.7, $\epsilon _{{\mathcal {M}}} \colon f^+f_*{\mathcal {M}} \to p^+_*p^*{\mathcal {M}}$ is an isomorphism, hence so is $p_*p^{+*}\epsilon _{{\mathcal {M}}}$.

Thus, for any ${\mathcal {M}} \in \mathscr {P}_{-1}$, we have an isomorphism $f^*f_* {\mathcal {M}} \to p_*p^{+*}p^+_*p^* {\mathcal {M}}$. By induction on triangles, the isomorphism extends to $\mathop {{\rm Hot}}\nolimits ^b(\mathscr {P}_{-1})$. Lemmas 4.9 and 4.11 imply that base change $\widetilde {\epsilon }$ is an isomorphism on ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$.

Proposition 4.19 Let $f\colon X \to Y$ satisfy assumption (a). The flop–flop functor $F^+F$ restricted to ${\mathcal {D}}^b(X)$ is isomorphic to term-wise functor $f^*f_*$ on the category ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$ under the equivalence ${\rm Hot}^{-,b}(\mathscr {P}_{-1})\simeq {\mathcal {D}}^b(X)$.

Proof. In view of Proposition 4.18, it suffices to check that functors $p_*p^{+*}p^+_*p^*$ and $Rp_*Lp^{+*}Rp^+_*Lp^*$ are isomorphic on ${\rm Hot}^{-,b}(\mathscr {P}_{-1})$. By Proposition 4.8, we have an isomorphism $Rp^+_* Lp^* {\mathcal {M}} \simeq p^+_* p^* {\mathcal {M}}$, for any ${\mathcal {M}} \in \mathscr {P}_{-1}$. Proposition 4.10 implies that $p^+_*p^* {\mathcal {M}}\simeq f^{+*}f_* {\mathcal {M}}$, hence $Rp_*Lp^{+*}p^+_*p^* {\mathcal {M}} \simeq p_* p^{+*}p^+_*p^* {\mathcal {M}}$ by Proposition 4.8. By induction on triangles, we have an isomorphism on $\mathop {{\rm Hot}}\nolimits ^b(\mathscr {P}_{-1})$. We conclude by Lemma 4.11.

4.5 The flop functor as the inverse of Van den Bergh's functor

We show that the flop functor $F^+\colon {\mathcal {D}}^b(X^+)\to {\mathcal {D}}^b(X)$ and functor $\Sigma$ are inverse to each other. In view of Theorem 4.5, this statement is equivalent to the flop functor $F^+$ being isomorphic to the term-wise functor $T = (f^*f^+_*(-))^{\vee \vee }$.

Proposition 4.20 Let $f\colon X \to Y$ satisfy assumption (a). The flop functor is isomorphic to $(f^{+*}f_*(-))^{\vee \vee }$ as a functor $\mathscr {P}_{0}\to \mathscr {P}_{-1}^+$.

Proof. Let ${\mathcal {N}}$ be an object in $\mathscr {P}_0$ and ${\mathcal {M}}^+ =(f^{+*}f_*{\mathcal {N}})^{\vee \vee }$ its image under the equivalences of Lemma 3.11. Lemma 2.9 and Proposition 4.18 give isomorphisms

\[ p^+_*p^{+*}f^{+*}f^+_* {\mathcal{M}}^+ \simeq p^+_*p^{+*}p^+_*p^{*}p_*p^{+*} {\mathcal{M}}^+ \simeq p^+_*p^{*}p_*p^{+*} {\mathcal{M}}^+ \simeq f^{+*}f^+_* {\mathcal{M}}^+. \]

Thus, sequence (20) implies that $Rp^+_*(L^1p^{+*}f^{+*}f^+_* {\mathcal {M}}^+) \simeq 0$. Moreover, we have $Rp^+_* p^{+*}{\mathcal {M}}^+ \simeq {\mathcal {M}}^+$. Hence, by applying $Rp^+_*$ to sequence (56), we obtain

\[ Rp^+_* p ^* {\mathcal{N}}\simeq Rp^+_* p^{+*}{\mathcal{M}}^+ \simeq {\mathcal{M}}^+. \]

Hence, the flop functor takes ${\mathcal {N}}$ to ${\mathcal {M}}^+$.

Let $\epsilon _{{\mathcal {N}}} \colon f^{+*}f_* {\mathcal {N}} \to p^+_*p^* {\mathcal {N}}\simeq {\mathcal {M}}^+$ be the base change. As $f_*{\mathcal {N}} \simeq f^+_* {\mathcal {M}}^+$, we have $f^{+*}f_* {\mathcal {N}} \simeq f^{+*}f^+_* {\mathcal {M}}^+$. The kernel of reflexification $f^{+*}f^+_* {\mathcal {M}}^+ \to (f^{+*}f^+_*{\mathcal {M}}^+)^{\vee \vee } \simeq {\mathcal {M}}^+$ is torsion and $p^{+*}p_* {\mathcal {N}} \simeq {\mathcal {M}}^+$ is torsion-free, hence, by applying functor $\mathop {{\rm Hom}}\nolimits _{X^+}(-, p^{+*}p_* {\mathcal {N}})$ to sequence (15) with $f$ and ${\mathcal {M}}$ replaced by $f^+$ and ${\mathcal {M}}^+$, we obtain an isomorphism $\mathop {{\rm Hom}}\nolimits _{X^+}({\mathcal {M}}^+, p^{+*}p_* {\mathcal {N}}) \simeq \mathop {{\rm Hom}}\nolimits _{X^+}(f^{+*}f^+_*{\mathcal {M}}^+, p^{+*}p_* {\mathcal {N}})$. It follows that $\epsilon _{{\mathcal {N}}}$ factors uniquely via a morphism $\widetilde {\epsilon }_{{\mathcal {N}}}\colon {\mathcal {M}}^+ \simeq (f^{+*}f_* {\mathcal {N}})^{\vee \vee } \to p^+_*p^*{\mathcal {N}}$ which is easily checked to be functorial. Thus, there exists a morphism $\widetilde {\epsilon }\colon (f^{+*}f_*(-))^{\vee \vee } \to p^+_*p^*(-)$ of functors.

Objects of $\mathscr {P}_0$ and $\mathscr {P}_{-1}^+$ are reflexive on $X$ and $X^+$, hence any morphism ${\mathcal {N}}_1 \to {\mathcal {N}}_2$ (and ${\mathcal {M}}^+_1 \to {\mathcal {M}}_2^+$) is determined by its restrictions to any open set with the complement of codimension greater than one. In particular, by its restriction to the complement $U \subset X$ of the exceptional set of $f$ (and $f^+$). Flop functor is $Rp^+_* Lp^*$ and the other functor is $(f^{+*}f_*(-))^{\vee \vee }$. Then, taking into account that maps $f$, $f^+$, $p$ and $p^+$ are isomorphisms outside the exceptional sets and $(f^{+})^{-1}\circ f=p^+\circ p^{-1}$ on $U$, the morphism $\mathop {{\rm Hom}}\nolimits _X({\mathcal {N}}_1, {\mathcal {N}}_2) \to \mathop {{\rm Hom}}\nolimits _{X^+}({\mathcal {M}}^+_1, {\mathcal {M}}^+_2)$ induced by the flop functor $F$ coincides with the morphism given by $(f^{+*}f_*(-))^{\vee \vee }$. It follows that $\widetilde {\epsilon }$ is an isomorphism of functors.

Theorem 4.21 Let $f\colon X\to Y$ satisfy assumption (a). The flop functor $F$ takes $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$ to $\mathop {{}^{-1}{\rm Per}}\nolimits (X^+/Y)$. It is isomorphic to $\Sigma _{{\rm qc}}^{-1}$ in the category of bimodule functors ${\mathcal {D}}_{{\rm qc}}(X)\to {\mathcal {D}}_{{\rm qc}}(X^+)$.

Proof. The same argument as in the proof of Proposition 4.6 shows that $\Sigma _{{\rm qc}}^{-1}$ is the term-wise extension of $(f^{+*}f_*(-))^{\vee \vee }$ to the category $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_0)$. Proposition 4.20 implies that $F|_{\mathscr {P}_0}\simeq \Sigma _{{\rm qc}}^{-1}|_{\mathscr {P}_0}$.

Categories ${\mathcal {D}}_{{\rm qc}}(X)$ and ${\mathcal {D}}_{{\rm qc}}(X^+)$ have compact generators ${\mathcal {N}}\in \mathscr {P}_0$ and ${\mathcal {M}}^+ = (f^{+*}f_*{\mathcal {N}})^{\vee \vee }\in \mathscr {P}_{-1}^+$. For algebras $A_X = \mathop {{\rm Hom}}\nolimits _X({\mathcal {N}}, {\mathcal {N}})$ and $A_{X^+} = \mathop {{\rm Hom}}\nolimits _{X^+}({\mathcal {M}}^+, {\mathcal {M}}^+)$, we have equivalences ${\mathcal {D}}_{{\rm qc}}(X) \simeq {\mathcal {D}}(\text{Mod--}A_X)$, ${\mathcal {D}}_{{\rm qc}}(X^+) \simeq {\mathcal {D}}(\text{Mod--}A_{X^+})$. By construction, functor $\Sigma _{{\rm qc}}^{-1}$ is isomorphic to a bimodule functor $\Phi _{M_1}$, for some $A_X^{\mathop {{\rm op}}\nolimits } \otimes A_{X^+}$ DG bimodule $M_1$, see Appendix C. Similarly, the flop functor $F$ is isomorphic to $\Phi _{M_2}$, for some DG bimodule $M_2$. Moreover, from the construction of $\Sigma _{{\rm qc}}^{-1}$, it follows that ${\mathcal {H}}^i(M_1)= 0$, for $i \neq 0$ (see the discussion before Proposition 4.6). By Lemma C.12, isomorphism $F|_{\mathscr {P}_0} \simeq \Sigma _{{\rm qc}}^{-1}|_{\mathscr {P}_0}$ implies an isomorphism $F\simeq \Sigma _{{\rm qc}}^{-1}$.

As an immediate consequence of Theorems 4.5 and 4.21 we obtain the following corollary.

Corollary 4.22 Let $f\colon X\to Y$ satisfy assumption (a). Then the flop functor $F$ induces an equivalence of ${\mathcal {D}}^b(X)$ with ${\mathcal {D}}^b(X^+)$.

In view of Proposition 4.13 and Theorem 4.21 we have a functorial exact triangle of functors ${\mathcal {D}}_{{\rm qc}}(X) \to {\mathcal {D}}_{{\rm qc}}(X^+)$

(64)\begin{equation} (F^+)^{-1}[1] \to Lg^{+*}Rg_* \to F \to (F^+)^{-1}[2]. \end{equation}

Moreover, it restricts to a functorial exact triangle of functors ${\mathcal {D}}^b(X)\to {\mathcal {D}}^b(X^+)$.

Corollary 4.23 Let $f\colon X\to Y$ satisfy assumption (a). The flop functor $F\colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X^+)$ is $t$-exact when ${\mathcal {D}}^b(X)$ is endowed with the $t$-structure with heart $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$ and ${\mathcal {D}}^b(X^+)$ with the $t$-structure with heart $\mathop {{}^{p-1}{\rm Per}}\nolimits (X^+/Y)$. Moreover, $F$ induces an equivalence of $\mathscr {A}_f$ with $\mathscr {A}_{f^+}[1]$.

Proof. As $Rf^+_* \circ F \simeq Rf_*$ and, by Theorem 4.21, functor $F$ takes $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$ to $\mathop {{}^{-1}{\rm Per}}\nolimits (X^+/Y)$, the flop $F$ induces an equivalence of $\mathscr {A}_f\subset \mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$ with $\mathscr {A}_{f^+}[1]\subset \mathop {{}^{-1}{\rm Per}}\nolimits (X^+/Y)$.

Category $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$ is defined by two conditions; we require that $Rf_* E$ is a pure sheaf, for any $E \in \mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$, and that $\mathscr {A}_f[-p] \subset \mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$. It follows from the properties of $F$ that $F(\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)) \subset \mathop {{}^{p-1}{\rm Per}}\nolimits (X^+/Y)$. As $F$ is an equivalence, both categories are hearts of bounded $t$-structures on ${\mathcal {D}}^b(X^+)$. It follows that $F(\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)) = \mathop {{}^{p-1}{\rm Per}}\nolimits (X^+/Y)$.

Corollary 4.24 Let morphism $f\colon X\to Y$ satisfy assumption (a). Then categories ${\mathcal {D}}^b(X)$ and ${\mathcal {D}}^b(\mathop {{}^{p}{\rm Per}}\nolimits (X/Y))$ are equivalent, for $p\in \mathbb {Z}$.

Proof. Corollary 4.23 implies that, for any $p \in \mathbb {Z}$, category $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$ is equivalent either to $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$ or to $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$. Hence, ${\mathcal {D}}^b(\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)) \simeq {\mathcal {D}}^b(\mathop {{}^{q}{\rm Per}}\nolimits (X/Y))$, for $q \in \{-1,0\}$, $q \cong p \, \mod 2$. By Proposition 4.4, the latter category is equivalent to ${\mathcal {D}}^b(X)$, which finishes the proof.

Let $f\colon X\to Y$ satisfy assumption (a). Proposition 2.11 implies that $Rp_* {\mathcal {O}}_{X\times _Y X^+} = {\mathcal {O}}_X$. Thus, commutativity of diagram (42) yields isomorphisms $F^+ \circ Lf^{+*} \simeq Rp_*Lp^{+*}Lf^{+*} \simeq Rp_*Lp^*Lf^* \simeq Lf^*$. Hence, also $F^+\circ Lg^{+*} \simeq Lg^*$. Composing triangle (64) with $F^+$ leads to a functorial exact triangle of functors ${\mathcal {D}}_{{\rm qc}}(X) \to {\mathcal {D}}_{{\rm qc}}(X)$:

(65)\begin{equation} \mathop{{\rm Id}}\nolimits_{{\mathcal{D}}_{{\rm qc}}(X)}[1] \to Lg^*Rg_* \to F^+F \to \mathop{{\rm Id}}\nolimits_{{\mathcal{D}}_{{\rm qc}}(X)}[2]. \end{equation}

4.6 Reduction to the affine case

We have shown with Corollary 4.22 that, if a morphism $f\colon X\to Y$ satisfies assumption (a), then the flop functor $F$ is an equivalence of ${\mathcal {D}}^b(X)$ with ${\mathcal {D}}^b(X^+)$. Now, following [Che02], we show that the flop functor is also an equivalence, provided morphism $f$ satisfies assumption (p).

Note that the condition on $Y$ to have hypersurface singularities means that completions of the local rings at all closed points of $Y$ are defined by one equation in regular complete local rings.

Lemma 4.25 Let $Y$ be an irreducible variety of dimension $n$ with hypersurface singularities. Then it has a finite open covering by affine subvarieties $Y_i$ such that every $Y_i$ admits a closed embedding as a principal divisor into a smooth affine ${\mathcal {Y}}_i$.

For the reader's convenience, we give the proof of this statement.

Proof. Let $y\in Y$ be a closed point. As Zariski topology is quasi-compact, it is enough to show the existence of an affine open neighbourhood of $y$ which admits a closed embedding into a smooth affine ${\mathcal {Y}}$ of dimension $n+1$. As the statement is local, we can assume that $Y$ is affine. Then $Y$ is realised as a closed subscheme in ${\mathbb {A}}^m$. We can assume that $m>n+1$, because otherwise the proof is obvious. Let ${\mathbb {A}}^m\subset {\mathbb {P}}^m$ be the compactification to a projective space and ${\bar {Y}}\subset {\mathbb {P}}^m$ the closure of $Y$ in ${\mathbb {P}}^m$.

As $Y$ has hypersurface singularities, the Zariski tangent space at point $y\in Y$ has dimension $\leqslant n+1$. It follows that we can choose a point $z$ in ${\mathbb {P}}^m\setminus {\bar {Y}}$ such that the line $(zy)\subset {\mathbb {P}}^m$ does not contain any point of ${\bar {Y}}$ except for $y$ and is not tangent to $Y$ at $y$.

Denote by $Y_1$ the image of ${\bar {Y}}$ under the projection $p:{\mathbb {P}}^m\to {\mathbb {P}}^{m-1}$ with centre in $z$. By the choice of $z$, the morphism ${\bar {Y}}\to Y_1$ is an isomorphism for some Zariski neighbourhoods of $y$ and $p(y)$. We can keep projecting in the same manner, if necessary, by replacing ${\bar {Y}}\subset {\mathbb {P}}^m$ with $Y_1\subset {\mathbb {P}}^{m-1}$, until we come to the projective space of dimension $n+1$ and a closed irreducible subvariety $Y'$ in it of dimension $n$ together with a morphism $\pi :{\bar {Y}}\to Y'$ which is an isomorphism in neighbourhoods of $y$ and $\pi (y)$.

We can find an affine open neighbourhood ${\mathcal {Y}} \subset {\mathbb {P}}^{n+1}$ of $\pi (y)$ such that $\pi$ gives an isomorphism of $Y'\cap {\mathcal {Y}}$ with a neighbourhood of $y\in Y$. By choosing a smaller affine ${\mathcal {Y}}$, if necessary, we can assume that $Y'\cap {\mathcal {Y}} \subset {\mathcal {Y}}$ is cut out by a single equation. Then $Y'\cap {\mathcal {Y}}$ is affine and embedded as a principal divisor in ${\mathcal {Y}}$, so we are done.

We consider $f\colon X\to Y$ which satisfies assumption (p). Lemma 4.25 implies the existence of a finite affine covering $Y = \bigcup Y_i$ such that, for every $i$, space $Y_i$ admits an embedding as a principal Cartier divisor to a smooth affine ${\mathcal {Y}}_i$.

For every $i$, we pull back the whole diagram to $Y_i$ as follows.

Note that, for any $i$, morphism $f_i$ satisfies assumption (a).

Proposition 4.26 Let $f\colon X\to Y$ and $f_i \colon X_i \to Y_i$ be as above. Assume that flop functors $F_i \colon {\mathcal {D}}^b(X_i) \to {\mathcal {D}}^b(X_i^+)$ are equivalences, for every $i$. Then the flop functor $F \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X^+)$ is an equivalence.

Proof. As $X$ is Gorenstein, category ${\mathcal {D}}^b(X)$ has a spanning class $\Omega = \{{\mathcal {O}}_{x} \}$, where $x$ runs over all closed points in $X$ (see [HLdS07, Lemma 1.26]). With such a choice of a spanning class the same argument as in [Che02, Proposition 3.2] proves that $F$ is an equivalence if all $F_i$ are.

5. Spherical pairs

With the notation of (42), we consider the category ${\mathcal {D}}^b(X\times _Y X^+)$ and its quotient by the intersection ${\mathcal {K}}^b$ of kernels of $Rp_*$ and $Rp^+_*$. We prove that ${\mathcal {D}}^b(\mathscr {A}_f)$ is equivalent to the full subcategory of ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$ defined as the kernel of $Rp^+_*$. We show that ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$ admits SODs $\langle {\mathcal {D}}^b(\mathscr {A}_f), {\mathcal {D}}^b(X^+)\rangle = \langle {\mathcal {D}}^b(X), {\mathcal {D}}^b(\mathscr {A}_f) \rangle$ and similarly for $\mathscr {A}_{f^+}$. We conclude that $({\mathcal {D}}^b(X), {\mathcal {D}}^b(X^+))$ and $({\mathcal {D}}^b(\mathscr {A}_f), {\mathcal {D}}^b(\mathscr {A}_{f^+}))$ are spherical pairs, which yields a geometric incarnation of the schober [KS14] related to the flop.

5.1 Category ${\mathcal {D}}(\mathscr {A}_f)$ as a full subcategory of ${\mathcal {D}}_{{\rm qc}}(X \times _Y X^+)$

Lemma 5.1 Let $f\colon X\to Y$ satisfy assumption (a) and let ${\mathcal {P}}$ be a projective object in $\mathscr {A}_f$. Then $p_*p^* {\mathcal {P}} \simeq {\mathcal {P}}$, $Rp_* L^1p^* {\mathcal {P}} = 0$ and $Rp_* p^* {\mathcal {P}} \simeq {\mathcal {P}}$.

Proof. By Corollary 4.23, the flop functor $F^+$ induces an equivalence of $\mathscr {A}_{f^+}$ and $\mathscr {A}_f[1]$. Hence, there exists projective ${\mathcal {P}}^+\in \mathscr {A}_{f^+}$ such that ${\mathcal {P}}[1] \simeq Rp_* Lp^{+*}{\mathcal {P}}^+$. As $L^ip^{+*} {\mathcal {P}}^+ = 0$, for $i>1$, by Lemma 4.15, and $Rp_*p^{+*}{\mathcal {P}}^+ = 0$, by Corollary 2.12, we have ${\mathcal {P}} \simeq p_* L^1p^{+*}{\mathcal {P}}^+$. Then, according to Lemma 2.9, we have $p_*p^* {\mathcal {P}} \simeq {\mathcal {P}}$ and $Rp_* L^1p^* {\mathcal {P}} = 0$. As $Rp_* Lp^* {\mathcal {P}} \simeq {\mathcal {P}}$, it follows that $Rp_* p^* {\mathcal {P}}\simeq {\mathcal {P}}$.

Lemma 5.2 Let $f\colon X \to Y$ satisfy assumption (a) and let ${\mathcal {M}}$ be a projective object in $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$. There exists an exact triangle

(66)\begin{equation} p^*{\mathcal{P}} \to Lp^{+*}(f^{+*}f_* {\mathcal{M}}) \to Lp^* {\mathcal{M}} \to p^*{\mathcal{P}}[1], \end{equation}

for ${\mathcal {P}} :={\mathcal {H}}_X^{-1}(\iota _f^* {\mathcal {M}})$ projective in $\mathscr {A}_f$.

Proof. By Lemma 4.15, we have $L^jp^* {\mathcal {P}} = 0$, for $j>1$ and any projective object ${\mathcal {P}}$ in $\mathscr {A}_f$. Thus, applying $Lp^*$ to sequence (15) and using the fact that ${\mathcal {M}}$ is locally free, we get an isomorphism $L^jp^* {\mathcal {P}} \simeq L^jp^*(f^*f_* {\mathcal {M}})$, for $j>0$. Further, as $Lp^*{\mathcal {M}} \simeq p^*{\mathcal {M}}$, i.e. $L^ip^*{\mathcal {M}} =0$, for $i\geqslant 1$, the sequence

(67)\begin{equation} 0\to p^* {\mathcal{P}} \to p^*f^*f_* {\mathcal{M}} \to p^* {\mathcal{M}}\to 0 \end{equation}

obtained by applying $p^*$ to sequence (15) is short exact. We also have

(68)\begin{equation} p^*f^*f_* {\mathcal{M}}\simeq Lp^{+*}(f^{+*}f^+_*{\mathcal{N}}^+) \simeq Lp^{+*}(f^{+*}f_*{\mathcal{M}}), \end{equation}

for ${\mathcal {N}}^+ \in \mathscr {P}^+_0$ given by (35). Indeed, commutativity of diagram (42) implies that $p^*f^*f_* {\mathcal {M}} \simeq p^{+*}f^{+*}f_* {\mathcal {M}}$. Further, Lemma 3.11 yields $f_* {\mathcal {M}} \simeq f^+_* {\mathcal {N}}^+$. Finally, ${\mathcal {P}}^+ = {\mathcal {H}}^0_X(\iota _{f^+}^*{\mathcal {N}}^+)$ is a projective object in $\mathscr {A}_{f^+}$, according to Lemma 3.5, hence $L^jp^{+*} {\mathcal {P}}^+ = 0$, for $j>1$ (by Lemma 4.15). Thus, applying $p^{+*}$ to sequence (32) on $X^+$ yields $L^jp^{+*}(f^{+*}f^+_* {\mathcal {N}}^+) = 0$, for $j>0$.

It follows that the triangle given by short exact sequence (67) is isomorphic to a triangle of the form (66).

Proposition 5.3 Let $f\colon X\to Y$ satisfy assumption (a) and let ${\mathcal {M}}$ be a projective object in $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$. Then $p^* {\mathcal {P}}\in \mathop {{\rm Perf}}\nolimits (X\times _Y X^+)$, $\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(p^* {\mathcal {P}}, p^* {\mathcal {P}}) \simeq \mathop {{\rm Hom}}\nolimits _X({\mathcal {P}}, {\mathcal {P}})$ and $\mathop {{\rm Ext}}\nolimits ^i_{X\times _Y X^+}(p^*{\mathcal {P}}, p^* {\mathcal {P}}) = 0$, for $i>0$, where ${\mathcal {P}} = {\mathcal {H}}_X^{-1}(\iota _f^* {\mathcal {M}})$ is projective in $\mathscr {A}_f$.

Proof. Sheaf ${\mathcal {M}}$ is locally free on $X$, hence $Lp^* {\mathcal {M}} \in \mathop {{\rm Perf}}\nolimits (X\times _Y X^+)$. Sequence (41) implies that $f^{+*}f^+_* {\mathcal {N}}^+ \in \mathop {{\rm Perf}}\nolimits (X^+)$, hence $Lp^{+*}f^{+*}f_* {\mathcal {N}}$ is also a perfect complex on $X\times _Y X^+$. Triangle (66) implies that $p^* {\mathcal {P}} \in \mathop {{\rm Perf}}\nolimits (X\times _Y X^+)$.

By adjunction and Lemma 5.1, we have $\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(p^* {\mathcal {P}},p^* {\mathcal {P}}) \simeq \mathop {{\rm Hom}}\nolimits _X({\mathcal {P}}, {\mathcal {P}})$. Thus, it suffices to check that $\mathop {{\rm Ext}}\nolimits ^i_{X\times _Y X^+}(p^* {\mathcal {P}}, p^* {\mathcal {P}}) = 0$, for $i>0$. Applying $\mathop {{\rm Hom}}\nolimits (-, p^* {\mathcal {P}})$ to triangle (66) yields, for any $i\in \mathbb {Z}$, an isomorphism

\[ \mathop{{\rm Ext}}\nolimits^{i+1}_{X\times_Y X^+}(Lp^* {\mathcal{M}}, p^* {\mathcal{P}}) \simeq \mathop{{\rm Ext}}\nolimits^i_{X\times_Y X^+}(p^*{\mathcal{P}}, p^* {\mathcal{P}}), \]

because $\mathop {{\rm Ext}}\nolimits ^j_{X\times _Y X^+}(Lp^{+*}(f^{+*}f_* {\mathcal {M}}), p^* {\mathcal {P}}) \simeq \mathop {{\rm Ext}}\nolimits ^j_{X^+}(f^{+*}f_* {\mathcal {M}}, Rp^+_* p^*{\mathcal {P}})= 0$ (see Corollary 2.12). Sheaf ${\mathcal {M}}$ is projective in $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ and ${\mathcal {P}}[1] \in \mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$. Thus, by Lemma 5.1 again, we conclude that

\[ \mathop{{\rm Ext}}\nolimits^{i+1}_{X\times_Y X^+}(Lp^* {\mathcal{M}}, p^* {\mathcal{P}}) \simeq \mathop{{\rm Ext}}\nolimits^{i+1}_X({\mathcal{M}}, Rp_*p^*{\mathcal{P}}) \simeq \mathop{{\rm Ext}}\nolimits^i_X({\mathcal{M}}, {\mathcal{P}}[1]) = 0, \]

for $i\neq 0$, which finishes the proof.

Now let ${\mathcal {M}}$ be a projective generator of $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ and ${\mathcal {P}}= {\mathcal {H}}_X^{-1}(\iota _f^* {\mathcal {M}})$. Denote by

\[ A_P := \mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(p^*{\mathcal{P}}, p^*{\mathcal{P}}) \]

the endomorphism algebra of $p^* {\mathcal {P}}$. Functor $\mathop {{\rm Hom}}\nolimits _{X\times _YX^+}(p^*{\mathcal {P}}, -)\colon {\rm QCoh}(X\times _YX^+) \to \text{Mod--}A_P$ has left adjoint $(-)\otimes _{A_P}p^*P\colon \text{Mod--}A_P \to {\rm QCoh}(X\times _Y X^+)$, [Pop73, Theorem 3.6.3]. We denote by $\bar {p}^* \colon {\mathcal {D}}(\mathscr {A}_f) \to {\mathcal {D}}_{{\rm qc}}(X\times _Y X^+)$ its derived functor.

As ${\mathcal {P}}$ is a projective generator of $\mathscr {A}_f$, categories ${\mathcal {D}}(\mathscr {A}_f)$ and ${\mathcal {D}}(\text{Mod--}A_P)$ are equivalent. We also have ${\mathcal {D}}(\text{Mod--}A_P) \simeq [\textit {SF--}A_P]$, for the category $\textit {SF--}A_P \subset \text{DGMod--}A_P$ of semi-free $A_P$ DG modules, i.e. DG modules that admit a filtration with direct sums of shifts of representable DG modules as the graded factors of the filtration [Dri04].

Proposition 5.4 Functor $\bar {p}^* \colon {\mathcal {D}}(\mathscr {A}_f) \to {\mathcal {D}}_{{\rm qc}}(X\times _Y X^+)$ is fully faithful.

Proof. By [Kel94, Lemma 4.2], in order to prove that functor ${\mathcal {D}}(\text{Mod--}A_P) \to {\mathcal {D}}_{qc}(X\times _Y X^+)$ is fully faithful, it suffices to show that it induces a bijection $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}(A_f)}({\mathcal {P}}, {\mathcal {P}}[n]) \to \mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(p^*{\mathcal {P}}, p^* {\mathcal {P}}[n])$, for any $n\in \mathbb {Z}$ and that $p^* {\mathcal {P}} \in {\mathcal {D}}_{{\rm qc}}(X\times _Y X^+)$ is compact. Both statements follow from Proposition 5.3.

Functor $\overline {p}^*$ restricts to a fully faithful functor ${\mathcal {D}}^b(\mathscr {A}_f)\to {\mathcal {D}}^-(X\times _Y X^+)$. We show that ${\mathcal {D}}^b(\mathscr {A}_f)$ is, in fact, a subquotient of ${\mathcal {D}}^b(X\times _Y X^+)$. Following an idea of Kapranov, we show that the flop–flop functor is a spherical cotwist arising from a spherical pair.

5.2 Spherical pairs $({\mathcal {D}}^b(X)$, ${\mathcal {D}}^b(X^+))$, $({\mathcal {D}}^b(\mathscr {A}_f), {\mathcal {D}}^b(\mathscr {A}_{f^+}))$

For a morphism $f\colon X\to Y$ satisfying (a) and its flop $f^+\colon X^+ \to Y$, we consider

(69)\begin{equation} {\mathcal{K}} = \{ E \in {\mathcal{D}}^-(X\times_Y X^+)\,|\, Rp_*(E) = 0,\, Rp^+_*(E) = 0\} \end{equation}

and quotient category ${\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$. In the above, as in diagram (42), morphisms $p$ and $p^+$ denote the projections for $X\times _Y X^+$. As both $p$ and $p^+$ have fibers of dimension bounded by one, category ${\mathcal {K}}$ inherits the standard $t$-structure from ${\mathcal {D}}^-(X\times _Y X^+)$ (Lemma 2.2). In particular, $K\in {\mathcal {K}}$ if and only if ${\mathcal {H}}^i(K) \in {\mathcal {K}}$, for all $i\in \mathbb {Z}$.

Let ${\rm Mor}_{{\mathcal {K}}}\subset {\mathcal {D}}^-(X\times _Y X^+)$ denote the category with objects as in ${\mathcal {D}}^-(X\times _Y X^+)$ and morphisms $f\colon E \to F$ such that the cone of $f$ lies in ${\mathcal {K}}$ (see [Ver77, Nee01]). The quotient category ${\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$ has the same objects as ${\mathcal {D}}^-(X\times _Y X^+)$. For any pair of objects $E, F$, morphisms $E\to F$ in ${\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$ are equivalence classes of triples $(f, Z, g)$, where $Z\xrightarrow {f} E$ is a morphism in ${\rm Mor}_{{\mathcal {K}}}$ and $Z\xrightarrow {g} F$ is a morphism in ${\mathcal {D}}^-(X\times _Y X^+)$. We have $(f,Z,g) \sim (f', Z', g')$ if there exists a triple $(f'', Z'', g'')$ and $Z''\xrightarrow {u} Z$, $Z''\xrightarrow {v} Z'$ for which diagram

(70)

commutes, cf. [Nee01].

We denote by ${\mathcal {K}}^b$ the intersection ${\mathcal {K}} \cap {\mathcal {D}}^b(X\times _Y X^+)$. The embedding ${\mathcal {D}}^b(X\times _Y X^+) \to {\mathcal {D}}^-(X\times _Y X^+)$ induces a functor

(71)\begin{equation} \chi \colon {\mathcal{D}}^b(X \times_Y X^+)/ {\mathcal{K}}^b \to {\mathcal{D}}^-(X\times_Y X^+)/{\mathcal{K}}. \end{equation}

Lemma 5.5 Functor $\chi$ is fully faithful.

Proof. It is clear from the above definitions that $\chi$ is well-defined. Let us check that $\chi$ is faithful. To this end, we assume that $(f,Z,g)$, $(f',Z',g')$ are morphisms $E_1 \to E_2$ in ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$ which are equivalent as morphisms in ${\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$. Let $(f'', Z'', g'')$ with $Z''\in {\mathcal {D}}^-(X\times _Y X^+)$ be the triple defining the equivalence as in (70). Choose $l\in \mathbb {Z}$ such that $E_1$, $E_2$, $Z$ and $Z'$ belong to ${\mathcal {D}}^b(X\times _Y X^+)^{\geqslant l}$. Then

(72)\begin{equation} {\mathcal{H}}^i({\rm Cone}(\tau_{\geqslant l} f'')) = \begin{cases} {\mathcal{H}}^i({\rm Cone}(f'')), & \textrm{for } i\geqslant l-1,\\ 0, & \textrm{for } i< l-1,\end{cases} \end{equation}

implies that all cohomology sheaves of ${\rm Cone}(\tau _{\geqslant l} f'')$ belong to ${\mathcal {K}}$, i.e. ${\rm Cone}(\tau _{\geqslant l}f'')$ is an object in ${\mathcal {K}}^b$. The truncation $\tau _{\geqslant l}$ of diagram (70) is isomorphic to the original diagram on the boundary diamond, whereas the middle line $(\tau _{\geqslant l}f'', \tau _{\geqslant l} Z'', \tau _{\geqslant l} g'')$ defines a morphism in ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$. Hence, $(f,Z,g) \sim (f', Z', g')$ holds in ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$.

Finally, let $(f, Z, g)$ be a morphism $E_1 \to E_2$ in ${\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$ and assume there exists $l\in \mathbb {Z}$ such that both ${\mathcal {H}}^i(E_1)$ and ${\mathcal {H}}^i(E_2)$ vanish, for $i\leqslant l$. Then (72) implies that $\tau _{\geqslant l}f \in {\rm Mor}_{{\mathcal {K}}}$. Hence, $(\tau _{\geqslant l} f, \tau _{\geqslant l} Z, \tau _{\geqslant l} g)$ is a morphism $E_1 \to E_2$ in ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$ which functor $\chi$ takes to a morphism equivalent to $(f,Z,g)$. Thus, functor $\chi$ is also full.

In view of Lemma 5.5, we can regard ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$ as a full subcategory of ${\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$.

Proposition 5.6 The composite of $Lp^* \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^-(X\times _Y X ^+)$ with the quotient functor $Q\colon {\mathcal {D}}^-(X\times _Y X^+) \to {\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$ takes ${\mathcal {D}}^b(X)$ to the essential image of $\chi$.

Proof. Let $E\in {\mathcal {D}}^b(X)^{\geqslant l}$. As $p$ has fibers of dimension bounded by one and $Rp_*Lp^*(E) \simeq E$, we have $Rp_* {\mathcal {H}}^j(Lp^* E) = 0$, for $j< l-1$.

As $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ is obtained from $\mathop {{\rm Coh}}\nolimits (X)$ by means of the tilt in a torsion pair, sequence (34) implies that ${\mathcal {H}}^j_{\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)}(E) =0$, for $j< l$. By Corollary 4.23, object $Rp^+_* Lp^*(E)$ lies in ${\mathcal {D}}^b(X^+)^{\geqslant l}_{\mathop {{}^{0}{\rm Per}}\nolimits (X^+/Y)}$. Category $\mathop {{\rm Coh}}\nolimits (X^+)$ is obtained from $\mathop {{}^{0}{\rm Per}}\nolimits (X^+/Y)$ by the tilt in a torsion pair, hence $Rp^+_*Lp^*E \in {\mathcal {D}}^b(X^+)_{X^+}^{\geqslant l-1}$. Finally, the dimension of fibers of $p^+$ being bounded by one implies that $Rp^+_* {\mathcal {H}}^j(Lp^* E) = 0$, for $j< l-2$. Thus, $\tau _{\leqslant l-3} Lp^*E$ lies in ${\mathcal {K}}$, i.e. morphism $Lp^*E \to \tau _{\geqslant l-2} Lp^* E$ is an isomorphism in ${\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$.

We denote by

\[ \widetilde{L}p^* \colon {\mathcal{D}}^b(X) \to {\mathcal{D}}^b(X\times_Y X^+)/{\mathcal{K}}^b \]

the functor invoked by Lemma 5.5 and Proposition 5.6.

In order to show existence of a right adjoint to $\widetilde {L}p^*$, we first prove the following general statement.

Lemma 5.7 Let ${\mathcal {D}}$ be a triangulated category and ${\mathcal {K}}$ a thick triangulated subcategory. Let further $E\in {\mathcal {D}}$ be such that $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}}(E, K) = 0$, for any $K\in {\mathcal {K}}$. Then, for any $F \in {\mathcal {D}}$, we have an isomorphism

\[ \mathop{{\rm Hom}}\nolimits_{{\mathcal{D}}}(E,F) \xrightarrow{\simeq} \mathop{{\rm Hom}}\nolimits_{{\mathcal{D}}/{\mathcal{K}}}(E,F). \]

Proof. We have the map

\[ \alpha \colon \mathop{{\rm Hom}}\nolimits_{{\mathcal{D}}}(E, F) \to \mathop{{\rm Hom}}\nolimits_{{\mathcal{D}}/{\mathcal{K}}}(E, F),\quad \alpha\colon h \mapsto ({\rm id}, E, h). \]

Now let $(f, Z, g)$ be any element in $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}/{\mathcal {K}}}(E, F)$. By definition, $f$ is a morphism in ${\rm Mor}_{{\mathcal {K}}}$. In other words we have an exact triangle

\[ Z \xrightarrow{f} E \rightarrow K \rightarrow Z[1], \]

with $K\in {\mathcal {K}}$. As both $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}}(E,K)$ and $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}}(E,K[-1])$ vanish, object $Z$ is uniquely decomposed as $Z \simeq K[-1] \oplus E$ and $f=(0, {\rm id})$ is the projection to $E$. Accordingly, we have a decomposition $g=(g_K,g_E)$. Diagram

commutes, i.e. $(f,Z,g) \sim ({\rm id}, E, g_E)$. Thus, map $\alpha$ is surjective.

Now let $({\rm id}, E, g) \sim ({\rm id}, E, g')$ in ${\mathcal {D}}/{\mathcal {K}}$. For the triple $(f'', Z'', g'')$ giving the equivalence (see diagram (70)), we have, as above, the decomposition $Z'' = K''[-1] \oplus E$. Commutativity of

yields $g = g''_E = g'$, i.e. $\alpha$ is also a monomorphism.

Proposition 5.8 Functor $\widetilde {L}p^*$ is fully faithful and has the right adjoint $Rp_* \colon {\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b\to {\mathcal {D}}^b(X)$.

Proof. First, note that functor $Rp_* \colon {\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b \to {\mathcal {D}}^b(X)$ is well-defined, because ${\mathcal {K}}^b\subset {\rm Ker} Rp_*$.

For any $E\in {\mathcal {D}}^b(X)$ and $F \in {\mathcal {D}}^-(X\times _Y X^+)$, we have an isomorphism $\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(Lp^* E, F) \simeq \mathop {{\rm Hom}}\nolimits _X(E, Rp_* F)$. As functor $\chi$ is fully faithful (Lemma 5.5), in order to prove that $Rp_*$ is right adjoint to $\widetilde {L}p^*$, it suffices to show that, for any $F \in {\mathcal {D}}^-(X\times _Y X^+)$, we have

(73)\begin{equation} \mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(Lp^* E, F) \simeq \mathop{{\rm Hom}}\nolimits_{{\mathcal{D}}^-(X\times_Y X^+)/{\mathcal{K}}}(Lp^*E, F). \end{equation}

By adjunction, $\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(Lp^*E, K) = 0$, for any $K\in {\mathcal {K}}$, thus (73) follows from Lemma 5.7.

As functor $Lp^* \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^-(X\times _Y X^+)$ is fully faithful, isomorphism (73), for $F = Lp^*(E')$, implies that functor $\widetilde {L}p^*$ is fully faithful too.

As $Rp_* \widetilde {L}p^* \simeq {\rm Id}_{{\mathcal {D}}^b(X)}$, we have a SOD [Bon89, Lemma 3.1]

(74)\begin{equation} {\mathcal{D}}^b(X\times_Y X^+)/{\mathcal{K}}^b = \langle {\mathcal{C}}, \widetilde{L}p^* {\mathcal{D}}^b(X)\rangle, \end{equation}

with ${\mathcal {C}} = \{ E\in {\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b\,|\, Rp_* E = 0\}$. We aim at showing that ${\mathcal {C}} \simeq {\mathcal {D}}^b(\mathscr {A}_{f^+})$.

By Proposition 5.3, we know that, for a projective object ${\mathcal {M}}\in \mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ and ${\mathcal {P}}= {\mathcal {H}}^{-1}_{X} \iota _{f}^* {\mathcal {M}}$, the endomorphisms algebras of $p^{*}{\mathcal {P}}$ and ${\mathcal {P}}$ are isomorphic. Next, we show that they remain the same when we pass to the quotient category ${\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$.

Lemma 5.9 Let ${\mathcal {P}}$ be as above. Then, for any $E\in {\mathcal {D}}^-(X\times _Y X^+)$,

\[ \mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(p^* {\mathcal{P}}, E) \simeq \mathop{{\rm Hom}}\nolimits_{{\mathcal{D}}^-(X\times_Y X^+)/{\mathcal{K}}}(p^*{\mathcal{P}}, E). \]

Proof. By Lemma 5.7, it suffices to check that $\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(p^*{\mathcal {P}}, K) = 0$, for any $K \in {\mathcal {K}}$. It immediately follows from applying $\mathop {{\rm Hom}}\nolimits (-, K)$ to triangle (66).

Consider the composite functor

\[ \widetilde{p}^* \colon {\mathcal{D}}^b(\mathscr{A}_f) \xrightarrow{\bar{p}^*|_{{\mathcal{D}}^b(\mathscr{A}_f)}} {\mathcal{D}}^-(X\times_Y X^+) \xrightarrow{Q} {\mathcal{D}}^-(X\times_Y X^+)/{\mathcal{K}}, \]

where $Q$ denotes the quotient functor and $\overline {p}^*$ is as in Proposition 5.4. Recall that we assume $Y= \mathop {{\rm Spec}}\nolimits R$ to be affine and Noetherian. Endomorphism algebra $A_P$ of ${\mathcal {P}}$ is finite over $R$, hence Noetherian. Thus, the category $\text{mod--}A_P$ of finitely generated right $A_P$ modules is abelian. As ${\mathcal {P}}$ is a projective generator of $\mathscr {A}_f$, ${\mathcal {D}}^b(\mathscr {A}_f)$ is equivalent to ${\mathcal {D}}^b(\text{mod--}A_P)$. Using this equivalence and taking into account Proposition 5.3, we can define $\widetilde {p}_*$, right adjoint to $\widetilde {p}^*$, as

(75)\begin{equation} \widetilde{p}_* \colon {\mathcal{D}}^b(X\times_Y X^+)\to {\mathcal{D}}^b(\text{mod--}A_P),\quad \widetilde{p}_*\colon E \mapsto R\mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(p^* {\mathcal{P}}, E). \end{equation}

Finally, let $\mathscr {P}$ be the category of projective objects in $\mathscr {A}_f$ of the form ${\mathcal {H}}^{-1}_X(\iota _f^* {\mathcal {M}})$, for ${\mathcal {M}}\in \mathscr {P}_{-1}$. As there is no non-trivial extension of objects in $\mathscr {P}$, it is an exact subcategory of $\mathscr {A}_f$. Moreover, for any $E \in \mathscr {A}_f$, there exists ${\mathcal {M}} \in \mathscr {P}_{-1}$ and $\varphi \colon {\mathcal {M}} \to E[1]$, surjective in $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$. As functor $\iota _{f*}\colon \mathscr {A}_f[1] \to \mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ is $t$-exact, its left adjoint $\iota _f^*$ is right exact, i.e. morphism ${\mathcal {H}}_X^{-1}(\iota _f^* {\mathcal {M}}) \to {\mathcal {H}}^{-1}_X(\iota _f^* E[1]) \simeq E$ is surjective in $\mathscr {A}_f$. This implies that

\[ {\mathcal{D}}^b(\mathscr{A}_f) \simeq \mathop{{\rm Hot}}\nolimits^{-,b}(\mathscr{P}). \]

The last equivalence allows us to view $\widetilde {p}^*$ as a functor

(76)\begin{equation} \widetilde{p}^{*}\colon \mathop{{\rm Hot}}\nolimits^{-,b}(\mathscr{P}) \to {\mathcal{D}}^-(X\times_Y X^+)/{\mathcal{K}}. \end{equation}

It is the term-wise extension of the functor $p^*\colon \mathscr {P}\to {\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$.

Lemma 5.1 implies that the adjunction unit $\mathop {{\rm Id}}\nolimits \to \widetilde {p}_* \widetilde {p}^*$ is an isomorphism on ${\mathcal {P}}$. By Lemma 4.11 we have an isomorphism

(77)\begin{equation} \mathop{{\rm Id}}\nolimits_{{\mathcal{D}}^b(\mathscr{A}_f)} \to \widetilde{p}_*\widetilde{p}^* \end{equation}

of functors ${\mathcal {D}}^b(\mathscr {A}_f) \to {\mathcal {D}}^b(\mathscr {A}_f)$.

The embedding $\mathscr {A}_f \to \mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$ is an exact functor of abelian categories. Hence, it induces the functor

\[ \Psi \colon {\mathcal{D}}(\mathscr{A}_f) \to {\mathcal{D}}_{{\rm qc}}(X). \]

Lemma 5.10 There exists an isomorphism $Rp_* \widetilde {p}^* \xrightarrow {\simeq } \Psi |_{{\mathcal {D}}^b(\mathscr {A}_f)}$ of functors $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}) \to {\mathcal {D}}^b(X)$.

Proof. Lemma 5.1 implies that the adjunction unit gives an isomorphism $\Psi |_{\mathscr {P}} \xrightarrow {\simeq } p_*p^*|_{\mathscr {P}}$ of functors $\mathscr {P} \to \mathop {{\rm Coh}}\nolimits (X)$. As functor $\Psi |_{{\mathcal {D}}^b(\mathscr {A}_f)}$ takes $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P})$ to ${\mathcal {D}}^b(X)$, Lemma 4.11 implies that the term-wise extension of $p_*p^*$ to the category $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P})$ is isomorphic to $\Psi |_{{\mathcal {D}}^b(\mathscr {A}_f)}$.

Now let ${\mathcal {P}}_{{\boldsymbol {\cdot }}}$ be a complex in $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P})$ and $F = \widetilde {p}^* {\mathcal {P}}_{{\boldsymbol {\cdot }}}$. Lemma 5.1 implies that the first layer of spectral sequence

\[ E_1^{q,r} = R^qp_* p^*{\mathcal{P}}_r \Rightarrow R^{q+r} p_* F. \]

has one non-zero row only. Hence, $Rp_* F$ is quasi-isomorphic to $p_* p^* {\mathcal {P}}_{{\boldsymbol {\cdot }}}$, which finishes the proof.

Proposition 5.11 There exists a fully faithful functor

\[ \widetilde{p}^* \colon {\mathcal{D}}^b(\mathscr{A}_f) \to {\mathcal{D}}^b(X\times_Y X^+)/{\mathcal{K}}^b. \]

Proof. Proposition 5.4 implies that $\widetilde {p}^* \colon {\mathcal {D}}^b(\mathscr {A}_f) \to {\mathcal {D}}^-(X\times _Y X^+)$ is fully faithful. In order to show that the composite ${\mathcal {D}}^b(\mathscr {A}_f) \xrightarrow {\widetilde {p}^*} {\mathcal {D}}^-(X\times _Y X^+) \xrightarrow {Q} {\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$ is fully faithful, we check that, for any $E \in {\mathcal {D}}^b(\mathscr {A}_f)$ and $F\in {\mathcal {D}}^-(X\times _Y X^+)$, we have

\[ \mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(\widetilde{p}^* E,F) \simeq \mathop{{\rm Hom}}\nolimits_{{\mathcal{D}}^-(X\times_Y X^+)/{\mathcal{K}}}(\widetilde{p}^* E,F). \]

By Lemma 5.7, it suffices to check that $\mathop {{\rm Hom}}\nolimits (\widetilde {p}^*E, K) = 0$, for any $E \in {\mathcal {D}}^b(\mathscr {A}_f)$ and any $K \in {\mathcal {K}}$. By induction on triangles, Lemma 5.9 implies that, for any $i\in \mathbb {Z}$, $\mathop {{\rm Hom}}\nolimits (\sigma _{\geqslant i} \widetilde {p}^*E, K) =0$. As, for any $P\in \mathscr {P}$, the space $\mathop {{\rm Hom}}\nolimits (p^*P, K)$ vanishes, we have $\mathop {{\rm Ext}}\nolimits ^j((\widetilde {p}^*E)^i,K) =0$, for any $i$ and $j$. It then follows from Lemma E.2 that $\mathop {{\rm Hom}}\nolimits (\widetilde {p}^*E, K) = 0$.

Thus, it suffices to show that functor $\widetilde {p}^*$ takes ${\mathcal {D}}^b(\mathscr {A}_f)$ to the image of ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$ in ${\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$, see (71). Consider $E\in {\mathcal {D}}^b({\mathcal {A}}_f)$. We shall show that $\widetilde {p}^* E$ is quasi-isomorphic in ${\mathcal {D}}^-(X\times _Y X^+)/{\mathcal {K}}$ to a complex with bounded cohomology. To this end, we shall find $N$ such that $\tau _{< N}\, \widetilde {p}^* E$ is an object in ${\mathcal {K}}$.

First, we note that $Rp^+_*(p^* {\mathcal {P}}_i) = 0$, for any $i\in \mathbb {Z}$ (Corollary 2.12). Thus, the term-wise definition of $\widetilde {p}^*$ (76) implies that $Rp^+_*(\widetilde {p}^* E) = 0$ and, as $p^+$ has fibers of dimension bounded by one, $Rp^+_* {\mathcal {H}}^i(\widetilde {p}^* E) =0$, for any $i\in \mathbb {Z}$ (see Lemma 2.2).

By Lemma 5.10, we have $Rp_* \widetilde {p}^* E \simeq \Psi (E) \in {\mathcal {D}}^b(X)^{\geqslant k}$, for some $k \in \mathbb {Z}$. As $p$ has fibers of dimension bounded by one, it follows that $Rp_* {\mathcal {H}}^i(\widetilde {p}^* E) = 0$, for $i< k-1$. Thus, $\tau _{< k-1} \widetilde {p}^* E$ lies in ${\mathcal {K}}$.

Lemma 5.12 Let ${\mathcal {M}}$ be a projective generator of $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ and put ${\mathcal {P}} = {\mathcal {H}}^{-1}_X(\iota _f^* {\mathcal {M}})$. Further, let $C\in {\mathcal {D}}^-(X\times _Y X^+)$ be such that $Rp^+_* C = 0$ and $R\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(p^*P, C) =0$. Then $Rp_* C = 0$.

Proof. Applying $R\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(-,C)$ to triangle (66) yields $R\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(Lp^*{\mathcal {M}}, C) \simeq R\mathop {{\rm Hom}}\nolimits _X({\mathcal {M}}, Rp_* C) =0$. As ${\mathcal {M}}$ is a compact generator of ${\mathcal {D}}_{{\rm qc}}(X)$, cf. [VdB04, Lemma 3.2.2] and the proof of Proposition 4.6, it follows that $Rp_* C = 0$.

Proposition 5.13 Category ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$ admits a SOD

\[ {\mathcal{D}}^b(X\times_Y X^+)/{\mathcal{K}}^b = \langle \widetilde{p}^* {\mathcal{D}}^b(\mathscr{A}_f), \widetilde{L}p^{+*} {\mathcal{D}}^b(X^+)\rangle. \]

Proof. As $\widetilde {p}^*$ is defined term-wise, Corollary 2.12 implies that $Rp^+_*\circ \widetilde {p}^* = 0$. In view of SOD (74) and Proposition 5.11, it suffices to show that any $E\in {\mathcal {D}}^b(X\times _YX^+)/{\mathcal {K}}^b$ such that $Rp^+_* E = 0$ is isomorphic to $\widetilde {p}^* F$, for some $F \in {\mathcal {D}}^b(\mathscr {A}_f)$.

Consider triangle

(78)\begin{equation} \widetilde{p}^*\widetilde{p}_*(E) \to E\to C_E \to \widetilde{p}^*\widetilde{p}_*(E)[1] \end{equation}

in ${\mathcal {D}}^-(X\times _Y X^+)$. As $Rp^+_* E = 0$ and $Rp^+_* \widetilde {p}^*(F)=0$, for any $F \in {\mathcal {D}}^b(\mathscr {A}_f)$, by applying $Rp^+_*$ to triangle (78) we obtain $Rp^+_* C_E =0$. In view of isomorphism (77), by applying $\widetilde {p}_*$ to triangle (78), we obtain $\widetilde {p}_* C_E = 0$. By formula (75), $R\mathop {{\rm Hom}}\nolimits (p^* {\mathcal {P}}, C_E) = 0$. By Lemma 5.12, $Rp_* C_E = 0$, i.e. $C_E \in {\mathcal {K}}$. Hence, $\widetilde {p}^*\widetilde {p}_*E\to E$ is an isomorphism in ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$.

The following corollary gives a geometric description for the category ${\mathcal {D}}^b(\mathscr {A}_f)$.

Corollary 5.14 We have an equivalence of categories

\[ {\mathcal{D}}^b(\mathscr{A}_f) = \{E \in {\mathcal{D}}^b(X\times_Y X^+)\, |\, Rp^+_*(E) = 0\}/{\mathcal{K}}^b. \]

Proposition 5.13 implies the existence of $\widetilde {p}_!\colon {\mathcal {D}}^b(X \times _Y X^+)/{\mathcal {K}}^b \to {\mathcal {D}}^b(\mathscr {A}_f)$, left adjoint to $\widetilde {p}^{*}$, which is defined by the functorial exact triangle

(79)\begin{equation} \widetilde{L}p^{+*} Rp^+_*\to \mathop{{\rm Id}}\nolimits \to \widetilde{p}^{*}\widetilde{p}_! \to \widetilde{L}p^{+*}Rp^+_*[1] \end{equation}

associated to a SOD. By exchanging the role of $X$ and $X^+$ we also obtain $\widetilde {p}^+_! \colon {\mathcal {D}}^b(X\times _YX^+)/{\mathcal {K}}^b \to {\mathcal {D}}^b(\mathscr {A}_{f^+})$.

Lemma 5.15 The composite $\widetilde {p}_!\widetilde {p}^{+*} \widetilde {p}^+_!\widetilde {p}^*\colon {\mathcal {D}}^b(\mathscr {A}_f)\to {\mathcal {D}}^b(\mathscr {A}_f)$ is isomorphic to $\mathop {{\rm Id}}\nolimits _{{\mathcal {D}}^b(\mathscr {A}_f)}[4]$.

Proof. First, we check that $\widetilde {p}^+_!\widetilde {p}^*$ restricts to a functor $\mathscr {P}\to \mathscr {P}^+$. By Lemma 5.10, $Rp_*\widetilde {p}^* \simeq \Psi$. Hence, for $P\in \mathscr {A}_f$ projective, triangle (79), with $p$ and $p^+$ interchanged, applied to $p^*P = \widetilde {p}^*P$ yields an exact triangle

\[ \widetilde{L}p^*P\to p^*P \to \widetilde{p}^{+*}\widetilde{p}^+_!\widetilde{p}^*P\to \widetilde{L}p^*P[1]. \]

By Lemma 4.15 object $Lp^*P$ has two non-zero cohomology sheaves, $p^*P$ and $L^1p^*P$. It follows that $\widetilde {p}^{+*}\widetilde {p}^+_!\widetilde {p}^*P\simeq L^1p^*P[2]$. Hence, in view of Lemma 5.10, $\Psi \widetilde {p}^+_!\widetilde {p}^*P \simeq Rp^+_*\widetilde {p}^{+*}\widetilde {p}^+_!\widetilde {p}^*P \simeq Rp^+_*L^1p^*P[2]$. As $Rp^+_*Lp^*$ restricts to an equivalence $\mathscr {A}_f\xrightarrow {\simeq } \mathscr {A}_{f^+}[1]$, Corollary 4.23, object $Rp^+_*L^1p^*P = P^+$ is projective in $\mathscr {A}_{f^+}$. Hence $\widetilde {p}^+_!p^*P[-2] = P^+$ is projective in $\mathscr {A}_{f^+}$.

As $\widetilde {p}^+_! \widetilde {p}^*[-2]$ maps objects of $\mathscr {P}$ to objects of $\mathscr {A}_{f^+}$, it maps acyclic complexes in $\mathscr {P}$ to acyclic complexes. Hence, the class $\mathscr {P}$ is adapted to $\widetilde {p}^+_!\widetilde {p}^*[-2]$, i.e. the functor can be calculated term-wise using the equivalence ${\mathcal {D}}^b(\mathscr {A}_f) \simeq \mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P})$. The same is true for the functor $\widetilde {p}_!\widetilde {p}^{+*}[-2]$ in the opposite direction. Moreover, because $\widetilde {p}^+_!\widetilde {p}^*[-2]$ maps complexes of objects of $\mathscr {P}$ to complexes of objects of $\mathscr {P}^+$ also the composite $\widetilde {p}_!\widetilde {p}^{+*}\widetilde {p}^+_!\widetilde {p}^*[-4]$ can be calculated term-wise. Therefore, it suffices to check that $\widetilde {p}_!\widetilde {p}^{+*}\widetilde {p}^+_!\widetilde {p}^*[-4]|_{\mathscr {P}}\simeq \mathop {{\rm Id}}\nolimits _{\mathscr {P}}$.

Precomposing triangle (79), with $p$ and $p^+$ interchanged, with $\widetilde {p}^*$, composing with $\widetilde {p}_!$ and using the isomorphism $\widetilde {p}_!\widetilde {p}^*\simeq \mathop {{\rm Id}}\nolimits$, we arrive at an exact triangle

(80)\begin{equation} \widetilde{p}_!\widetilde{L}p^*Rp_*p^*P \to P \to \widetilde{p}_!\widetilde{p}^{+*} \widetilde{p}^+_!\widetilde{p}^*P \to \widetilde{p}_!\widetilde{L}p^*Rp_*p^*P[1]. \end{equation}

First, we calculate the image of $\widetilde {p}_!\widetilde {L}p^*Rp_*p^*P \simeq \widetilde {p}_!\widetilde {L}p^*\Psi P \simeq \widetilde {p}_!\widetilde {L}p^*P$ under $\Psi = Rp_*\widetilde {p}^*$ using triangle (79) composed with $Rp_*$ and applied to $\widetilde {L}p^*P$:

\[ Rp_*Lp^{+*}Rp^+_*\widetilde{L}p^*P \to Rp_* \widetilde{L}p^*P \to \Psi\widetilde{p}_!\widetilde{L}p^*P \to Rp_*Lp^{+*}Rp^+_*\widetilde{L}p^*P [1]. \]

By Lemma 5.1 we have $Rp_*\widetilde {L}p^*P \simeq P$. As $Rg_*P =0$, triangle (65) implies that $Rp_*Lp^{+*}Rp^+_*\widetilde {L}p^*P \simeq P[2]$. It follows that $\Psi \widetilde {p}_!\widetilde {L}p^*P$ has non-zero cohomology in degrees $0$ and $-3$ only, both isomorphic to $P$. The map $\widetilde {p}_!\widetilde {L}p^*Rp_*p^*P \to P$ in (80) is the composition of two adjunction counits $\widetilde {L}p^*Rp_*\to \mathop {{\rm Id}}\nolimits$ and $\widetilde {p}_!\widetilde {p}^* \to \mathop {{\rm Id}}\nolimits$. The first when applied to $p^*P$ is the truncation at the zero cohomology, the second is an isomorphism. Hence, $\widetilde {p}_!\widetilde {p}^{+*} \widetilde {p}^+_!\widetilde {p}^*P \simeq P[4]$.

Proposition 5.16 Category ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$ admits a SOD

(81)\begin{equation} {\mathcal{D}}^b(X\times_Y X^+)/{\mathcal{K}}^b \simeq \langle \widetilde{L}p^*{\mathcal{D}}^b(X), \widetilde{p}^*{\mathcal{D}}^b(\mathscr{A}_f)\rangle. \end{equation}

Proof. Functor $\widetilde {p}_*(-) \simeq R\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b}(p^*{\mathcal {P}}, -)$ is right adjoint to the fully faithful functor $\widetilde {p}^*$ of Proposition 5.11.

First, we check that $\widetilde {p}_*\widetilde {L}p^*(E) = 0$, for any $E \in {\mathcal {D}}^b(X)$. It follows from Lemma 5.9 that the above statement is equivalent to

\[ \mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(p^*{\mathcal{P}}, Lp^*E) = 0, \]

for any $E \in {\mathcal {D}}^b(X)$.

By applying $\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(-, Lp^*E)$ to triangle (66), we learn that $\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(p^* {\mathcal {P}}, Lp^* E)$ is a cone of a morphism

\[ \theta_E\colon \mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(Lp^*{\mathcal{M}}, Lp^*E) \to \mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(Lp^{+*}(f^{+*}f_*{\mathcal{M}}), Lp^*E). \]

As $Rp_*{\mathcal {O}}_{X\times _Y X^+} \simeq {\mathcal {O}}_X$, we have

(82)\begin{equation} \mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(Lp^*{\mathcal{M}}, Lp^*E)\simeq \mathop{{\rm Hom}}\nolimits_X({\mathcal{M}}, E). \end{equation}

Moreover, in view of Proposition 4.10, we have

(83)\begin{equation} \mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(Lp^{+*}f^{+*}f_*{\mathcal{M}}, Lp^*E)\simeq \mathop{{\rm Hom}}\nolimits_{X^+}(F({\mathcal{M}}), F(E)). \end{equation}

We check that under isomorphisms (82) and (83) morphism $\theta _E$ coincides with the morphism $\nu _E$ induced by functor $F$:

\[ \nu_E\colon \mathop{{\rm Hom}}\nolimits_X({\mathcal{M}}, E) \to \mathop{{\rm Hom}}\nolimits_{X^+}(F({\mathcal{M}}), F(E)). \]

By (67) and (68), morphism $\varphi \colon Lp^{+*}f^{+*}f_*{\mathcal {M}}\to Lp^*{\mathcal {M}}$ in triangle (66) is the composite of the isomorphism $\kappa \colon Lp^{+*}f^{+*}f_*{\mathcal {M}} \xrightarrow {\simeq } p^{+*}f^{+*}f_*{\mathcal {M}} \xrightarrow {\simeq } p^*f^*f_*{\mathcal {M}}$ with the (non-derived) pull-back $p^*$ of the $f^*\dashv f_*$ adjunction counit $\varepsilon _{{\mathcal {M}}}\colon f^*f_*{\mathcal {M}}\to {\mathcal {M}}$. We use isomorphism (82) to identify $\alpha \in \mathop {{\rm Hom}}\nolimits _X({\mathcal {M}},E)$ with $Lp^*(\alpha )\in \mathop {{\rm Hom}}\nolimits (Lp^*({\mathcal {M}}), Lp^*(E))$. Under isomorphism (83), the composition $Lp^*(\alpha ) \circ \varphi$ reads as $Rp^+_*Lp^*(\alpha ) \circ Rp^+_*(p^*(\varepsilon _{{\mathcal {M}}})\circ \kappa ) \circ \eta _{{\mathcal {M}}}$, for the $Lp^{+*} \dashv Rp^+_*$ adjunction unit $\eta _{{\mathcal {M}}} \colon f^{+*}f_*{\mathcal {M}} \to Rp^+_*Lp^{+*}f^{+*}f_*{\mathcal {M}}$. The composite $f^{+*}f_*{\mathcal {M}} \xrightarrow {\eta _{{\mathcal {M}}}} Rp^+_*Lp^{+*}f^{+*}f_*{\mathcal {M}} \xrightarrow {Rp^+_*(\kappa )} Rp^+_*p^*f^*f_*{\mathcal {M}} \xrightarrow {Rp^+_*(\varepsilon _{{\mathcal {M}}})} Rp^+_*p^*{\mathcal {M}}$ is the base change (49). It follows that $Rp^+_*Lp^*(\alpha ) \circ Rp^+_*(p^*(\varepsilon _{{\mathcal {M}}})\circ \kappa ) \circ \eta _{{\mathcal {M}}}$ is the image of $F(\alpha ) = Rp^+_*Lp^*(\alpha ) \in \mathop {{\rm Hom}}\nolimits (F({\mathcal {M}}), F(E))$ under the isomorphism $\mathop {{\rm Hom}}\nolimits (F({\mathcal {M}}), F(E)) \simeq \mathop {{\rm Hom}}\nolimits (f^{+*}f_*{\mathcal {M}}, F(E))$ given by Proposition 4.10.

As the flop functor is an equivalence of categories (see Corollary 4.22), $\nu _E$ is an isomorphism, i.e. $\mathop {{\rm Hom}}\nolimits _{X \times _Y X^+}(p^*{\mathcal {P}}, Lp^*E) \simeq {\rm Cone}(\theta _E) \simeq 0$.

In order to prove the SOD (81) it suffices to show that, for any $E\in {\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$ such that $\widetilde {p}_*(E)\simeq 0$, we have $E\simeq \widetilde {L}p^*(E')$, for some $E'\in {\mathcal {D}}^b(X)$.

Now let $E\in {\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$ satisfy $\widetilde {p}_*E \simeq 0$. Consider an exact triangle

(84)\begin{equation} \widetilde{L}p^*Rp_* E\to E \to \widetilde{C}_E \to \widetilde{L}p^*Rp_* E [1]. \end{equation}

We have $Rp_* \widetilde {C}_E =0$. Moreover, because $\widetilde {p}_* \widetilde {L}p^*Rp_*E = 0$, it follows that $\widetilde {p}_*C_E=0$. Now, we proceed as in the proof of Lemma 5.12 to show that $Rp^+_*\widetilde {C}_E \simeq 0$.

Applying $R\mathop {{\rm Hom}}\nolimits _{X\times _Y X^+}(-,\widetilde {C}_E)$ to triangle (66) yields

\[ R\mathop{{\rm Hom}}\nolimits_{X\times_Y X^+}(Lp^{+*}(f^{+*}f_*{\mathcal{M}}), \widetilde{C}_E) \simeq R\mathop{{\rm Hom}}\nolimits_{X^+}(f^{+*}f_*{\mathcal{M}}, Rp^+_* \widetilde{C}_E) =0. \]

By Proposition 4.10, the sheaf $f^{+*}f_*{\mathcal {M}}$ is isomorphic to the flop $F({\mathcal {M}})$. As the flop functor $F$ is an equivalence (see Corollary 4.22), $F({\mathcal {M}})$ is a compact generator of ${\mathcal {D}}^b(X^+)$. Thus, vanishing of $R\mathop {{\rm Hom}}\nolimits _{X^+}(F({\mathcal {M}}), Rp^+_*\widetilde {C}_E)$ implies that $Rp^+_*\widetilde {C}_E \simeq 0$.

Hence, in triangle (84) sheaf $\widetilde {C}_E$ is an object of ${\mathcal {K}}^b$. Therefore, $E \simeq \widetilde {L}p^*Rp_* E$ in ${\mathcal {D}}^b(X\times _Y X^+)/{\mathcal {K}}^b$.

Theorem 5.17 Category ${\mathcal {D}}^b(X\times _YX^+)/{\mathcal {K}}^b$ admits 4-periodical SODs

\begin{align*} {\mathcal{D}}^b(X\times_YX^+)/{\mathcal{K}}^b &= \langle \widetilde{p}^*{\mathcal{D}}^b(\mathscr{A}_f), \widetilde{L}p^{+*}{\mathcal{D}}^b(X^+)\rangle = \langle \widetilde{L}p^{+*}{\mathcal{D}}^b(X^+), \widetilde{p}^{+*}{\mathcal{D}}^b(\mathscr{A}_{f^+})\rangle \\ &= \langle \widetilde{p}^{+*}{\mathcal{D}}^b(\mathscr{A}_{f^+}), \widetilde{L}p^{*}{\mathcal{D}}^b(X)\rangle = \langle \widetilde{L}p^{*}{\mathcal{D}}^b(X), \widetilde{p}^{*}{\mathcal{D}}^b(\mathscr{A}_{f})\rangle. \end{align*}

In particular, $(\widetilde {p}^*{\mathcal {D}}^b(\mathscr {A}_f), \widetilde {p}^{+*}{\mathcal {D}}^b(\mathscr {A}_{f^+}))$ and $(\widetilde {L}p^*{\mathcal {D}}^b(X), \widetilde {L}p^{*}{\mathcal {D}}^b(X^+))$ are spherical pairs.

Proof. Propositions 5.13 and 5.16 for $X$ and $X^+$ imply the 4-periodical SODs. Proposition B.3 claims that $(\widetilde {p}^*{\mathcal {D}}^b(\mathscr {A}_f), \widetilde {p}^{+*}{\mathcal {D}}^b(\mathscr {A}_{f^+}))$ and $(\widetilde {L}p^*{\mathcal {D}}^b(X), \widetilde {L}p^{*}{\mathcal {D}}^b(X^+))$ are spherical pairs.

Corollary 5.18 Functor $\Psi \colon {\mathcal {D}}^b(\mathscr {A}_f) \to {\mathcal {D}}^b(X)$ is spherical. The unit and counit for the $\Psi ^* \dashv \Psi$ adjunction fit into functorial exact triangles:

(85)\begin{equation} \Psi^*\Psi \to \mathop{{\rm Id}}\nolimits_{{\mathcal{D}}^b(\mathscr{A}_f)} \to \mathop{{\rm Id}}\nolimits_{{\mathcal{D}}^b(\mathscr{A}_f)}[4] \to \Psi^*\Psi[1] , \quad F^+F \to \mathop{{\rm Id}}\nolimits_{{\mathcal{D}}^b(X)} \to \Psi \Psi^* \to {F^+F}[1]. \end{equation}

Proof. By Proposition B.3 and Theorem 5.17 $(\widetilde {p}^*{\mathcal {D}}^b(\mathscr {A}_f), \widetilde {p}^{+*}{\mathcal {D}}^b(\mathscr {A}_{f^+}))$ is a spherical pair. By Proposition B.2 the corresponding spherical functor is $Rp_*\widetilde {p}^*$. By Lemma 5.10 the latter is isomorphic to $\Psi$.

By Proposition B.2 the cone of the counit $\Psi ^*\Psi \to \mathop {{\rm Id}}\nolimits _{{\mathcal {D}}^b(\mathscr {A}_f)}$ is isomorphic to $\widetilde {p}_!\widetilde {p}^{+*}\widetilde {p}^+_!\widetilde {p}^*$ and the cone of the unit $\mathop {{\rm Id}}\nolimits _{{\mathcal {D}}^b(X)} \to \Psi \Psi ^*$ is $Rp_*\widetilde {L}p^{+*}Rp^+_*\widetilde {L}p^*\simeq Rp_*Lp^{+*}Rp^+_*Lp^*=F^+F$. We conclude by the isomorphism $\widetilde {p}_!\widetilde {p}^{+*}\widetilde {p}^+_!\widetilde {p}^* \simeq \mathop {{\rm Id}}\nolimits _{{\mathcal {D}}^b(\mathscr {A}_f)}[4]$, Lemma 5.15.

6. Contraction algebra as the endomorphism algebra of a projective generator

In [DW16] Donovan and Wemyss considered a morphism $f\colon X\to Y$ between Gorenstein varieties of dimension three that contracts a rational irreducible curve $C$ to a point. If $Y$ is a spectrum of a complete local Noetherian ring, the category $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ has projective generator. In [DW16] a contraction algebra $A_{\rm con}$ is defined as the quotient of the endomorphism algebra of the projective generator for $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$ by morphisms which factor through ${\mathcal {O}}_X$. Algebra $A_{\rm con}$ is shown to govern non-commutative deformations of ${\mathcal {O}}_C(-1)$. If $N_{X/C} \simeq {\mathcal {O}}_C\oplus {\mathcal {O}}_C(-2)$, algebra $A_{\rm con}$ is commutative and was first considered in terms of deformation theory by Toda in [Tod07].

We prove in our, more general, set-up that $A_{\rm con}$ is isomorphic to the endomorphism algebra of a projective generator for $\mathscr {A}_f$.

Category $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$ has a projective generator ${\mathcal {N}}$ and $Q = {\mathcal {H}}^0_X(\iota _f^*{\mathcal {N}})$ is a projective generator for $\mathscr {A}_f$, Lemma 3.5. Endomorphisms algebras $A_Q = \mathop {{\rm End}}\nolimits _X({\mathcal {Q}})$ and $A_X = \mathop {{\rm End}}\nolimits _X({\mathcal {N}})$ are Noetherian, and we have equivalences ${\mathcal {D}}(\text{Mod--}A_Q)\simeq {\mathcal {D}}(\mathscr {A}_f)$, ${\mathcal {D}}(\text{Mod--}A_X)\simeq {\mathcal {D}}_{{\rm qc}}(X)$.

Projective generator ${\mathcal {Q}}$ for $\mathscr {A}_f$ is an object in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$. It follows that ${\mathcal {Q}}' :=R\mathop {{\rm Hom}}\nolimits _X({\mathcal {N}},{\mathcal {Q}})$ is an $A_Q^{\mathop {{\rm op}}\nolimits }\otimes _k A_X$ module and, under the above equivalences, functor $\Psi$ is isomorphic to

\[ \Psi\simeq \Phi_{Q'} \colon {\mathcal{D}}(\text{Mod--}A_Q)\to {\mathcal{D}}(\text{Mod--}A_X). \]

Proposition 3.19 yields a finite projective resolution of ${\mathcal {Q}}$ in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$. Hence, ${\mathcal {Q}}'$ is a perfect $A_X$ DG module and by Proposition C.6 the right adjoint $\Psi ^!$ exists and is isomorphic to a bimodule functor.

Proposition 6.1 Let $f$ satisfy assumption (a), let ${\mathcal {M}}_i$ be as in (17) and let ${\mathcal {P}}_i$ be as in (19). Then $\Psi \Psi ^!({\mathcal {M}}_i)\simeq \Psi {\mathcal {P}}_i[-2]$.

Proof. Considering the right adjoint to the functorial exact triangles of Corollary 5.18 yields an exact triangle:

(86)\begin{equation} \Psi \Psi^!({\mathcal{M}}_i) \to {\mathcal{M}}_i \to (F^+F)^{-1}({\mathcal{M}}_i)\to \Psi \Psi^!({\mathcal{M}}_i)[1]. \end{equation}

Put ${\mathcal {K}}_i=(F^+F)^{-1}({\mathcal {M}}_i)$. As $Rf_* F^+F = Rf_*$, we have $Rg_*{\mathcal {K}}_i \simeq Rg_*{\mathcal {M}}_i$. Hence, by Proposition 3.15, ${\mathcal {H}}^0_X(Lg^*Rg_*{\mathcal {K}}_i)=f^*f_*{\mathcal {M}}_i$, ${\mathcal {H}}^{-1}_X(Lg^*Rg_*{\mathcal {K}}_i) = {\mathcal {M}}_i$ and the remaining cohomology sheaves vanish. Considering long exact sequence of cohomology sheaves of the exact triangle obtained by applying functorial exact triangle (65) to ${\mathcal {K}}_i$:

\[ {\mathcal{K}}_i[1] \to Lg^*Rg_*{\mathcal{K}}_i \to {\mathcal{M}}_i \to {\mathcal{K}}_i[2], \]

we get that ${\mathcal {H}}^0({\mathcal {K}}_i) \simeq {\mathcal {M}}_i$ and ${\mathcal {H}}^1({\mathcal {K}}_i) \simeq {\mathcal {P}}_i$ (as ${\mathcal {P}}_i$ is the kernel of the surjective morphism $f^*f_*{\mathcal {M}}_i \to {\mathcal {M}}_i$). The long exact sequence of cohomology sheaves of (86) yields an isomorphism $\Psi \Psi ^!({\mathcal {M}}_i) \simeq {\mathcal {P}}_i[-2]\simeq \Psi {\mathcal {P}}_i[-2]$.

Theorem 6.2 Let $f$ satisfy assumption (c), let ${\mathcal {M}}_i$ be as in (17) and let ${\mathcal {P}}_i$ be as in (19). The endomorphism algebra

\[ A_P= \mathop{{\rm Hom}}\nolimits_{\mathscr{A}_f}\bigg(\bigoplus_{i=1}^n {\mathcal{P}}_i, \bigoplus_{i=1}^n {\mathcal{P}}_i\bigg) \]

is isomorphic to the quotient of $\mathop {{\rm Hom}}\nolimits _X(\bigoplus _{i=1}^n{\mathcal {M}}_i, \bigoplus _{i=1}^n {\mathcal {M}}_i)$ by the subspace of morphisms that factor via direct sums of copies of the structure sheaf.

Proof. By Proposition 6.1 $\Psi \Psi ^!({\mathcal {M}}_i) \simeq \Psi {\mathcal {P}}_i[-2]$. As $\Psi |_{\mathscr {A}_f}$ is fully faithful, we have $\Psi ^!({\mathcal {M}}_i) \simeq {\mathcal {P}}_i[-2]$. Thus, functor $\Psi ^!$ gives a homomorphism of algebras

\[ \varphi\colon \mathop{{\rm Hom}}\nolimits_X({\mathcal{M}}, {\mathcal{M}}) \to A_P = \mathop{{\rm Hom}}\nolimits_{{\mathcal{D}}^b(\mathscr{A}_f)}(\Psi^! {\mathcal{M}}, \Psi^! {\mathcal{M}}) \simeq \mathop{{\rm Hom}}\nolimits_X(\Psi \Psi^! {\mathcal{M}}, {\mathcal{M}}), \]

where ${\mathcal {M}} = \bigoplus _{i} {\mathcal {M}}_i$. The triangle of functors right adjoint to the second triangle of (85) implies that the morphism $\varphi$ fits into an exact sequence

\begin{align*} &\mathop{{\rm Hom}}\nolimits_X((F^+F)^{-1}{\mathcal{M}}, {\mathcal{M}}) \xrightarrow{\psi} \mathop{{\rm Hom}}\nolimits_X({\mathcal{M}}, {\mathcal{M}})\xrightarrow{\varphi} \mathop{{\rm Hom}}\nolimits_X(\Psi \Psi^! {\mathcal{M}}, {\mathcal{M}}) \\ &\quad \to \mathop{{\rm Hom}}\nolimits_X((F^+F)^{-1}{\mathcal{M}}[-1], {\mathcal{M}}). \end{align*}

By Corollary 4.22,

\[ \mathop{{\rm Hom}}\nolimits_X(F^{-1} {F^+}^{-1}({\mathcal{M}})[-1], {\mathcal{M}}) \simeq \mathop{{\rm Ext}}\nolimits^1_{X^+}({F^+}^{-1}({\mathcal{M}}), F({\mathcal{M}})). \]

Theorem 4.21 implies that ${F^+}^{-1}({\mathcal {M}}) \simeq {\mathcal {N}}^+$ is a projective object in $\mathop {{}^{0}{\rm Per}}\nolimits (X^+/Y)$. On the other hand, $F({\mathcal {M}}) \simeq f^{+*}f_* {\mathcal {M}}$ by Proposition 4.19. By Lemma 3.11, $f^{+*}f_*{\mathcal {M}} \simeq f^{+*}f^+_*{\mathcal {N}}^+$. Then, Lemma 2.9 implies that $R^1f^+_*(f^{+*}f_* {\mathcal {M}}) = 0$, i.e. $f^{+*}f_* {\mathcal {M}}$ is an object in $\mathop {{}^{0}{\rm Per}}\nolimits (X^+/Y)$ (see Remark 2.3 and formula (8) therein). Thus,

\[ \mathop{{\rm Hom}}\nolimits_X((F^+F)^{-1} {\mathcal{M}}[-1], {\mathcal{M}}) \simeq \mathop{{\rm Ext}}\nolimits^1_{X^+}({\mathcal{N}}^+, f^{+*}f_* {\mathcal{M}}) \simeq \mathop{{\rm Ext}}\nolimits^1_{\mathop{{}^{0}{\rm Per}}\nolimits(X/Y^+)}({\mathcal{N}}^+, f^{+*}f_* {\mathcal{M}})= 0, \]

i.e. morphism $\varphi$ is surjective.

Let us now describe the image of $\psi$. Triangle (64) with roles of $X$ and $X^+$ interchanged, when applied to ${\mathcal {N}}^+$, gives an exact triangle

\[ {\mathcal{M}}[-2] \to (F^+F)^{-1}{\mathcal{M}} \to Lg^*Rg_* {\mathcal{M}}[-1] \to {\mathcal{M}}[-1], \]

in view of $(F^+F)^{-1} {\mathcal {M}} \simeq F^{-1}({\mathcal {N}}^+)$, $Rg^+_* {\mathcal {N}}^+ \simeq Rg_* {\mathcal {M}}$, and $F^+({\mathcal {N}}^+)\simeq {\mathcal {M}}$. As $\mathop {{\rm Ext}}\nolimits ^1_X({\mathcal {M}}, {\mathcal {M}})= 0 = \mathop {{\rm Ext}}\nolimits ^2_X({\mathcal {M}}, {\mathcal {M}})$, we have isomorphisms $\mathop {{\rm Hom}}\nolimits _X({\mathcal {M}}, (F^+F)^{-1}{\mathcal {M}}) \simeq \mathop {{\rm Hom}}\nolimits _X({\mathcal {M}}, Lg^*Rg_* {\mathcal {M}}[-1])$ and $\mathop {{\rm Hom}}\nolimits _X((F^+F)^{-1}{\mathcal {M}}, {\mathcal {M}}) \simeq \mathop {{\rm Hom}}\nolimits _X(Lg^*Rg_* {\mathcal {M}}[-1], {\mathcal {M}})$. Thus, a morphism ${\mathcal {M}}\to {\mathcal {M}}$ is decomposed as ${\mathcal {M}} \to (F^+F)^{-1}{\mathcal {M}} \to {\mathcal {M}}$ if an only if it is decomposed as ${\mathcal {M}} \to Lg^*Rg_* {\mathcal {M}}[-1] \to {\mathcal {M}}$. By Lemma 3.14, object $Lg^*Rg_* {\mathcal {M}}$ is quasi-isomorphic to a complex $\{{\mathcal {O}}_X^S \to {\mathcal {O}}_X^S\}$, for some $S\in \mathbb {Z}_+$. Thus, ‘stupid’ truncation gives an exact triangle

\[ {\mathcal{O}}_X^S[-2] \to {\mathcal{O}}_X^S \to Lg^*Rg_* {\mathcal{M}}[-1] \to {\mathcal{O}}_X^S[-1]. \]

As $\mathop {{\rm Ext}}\nolimits ^1_X({\mathcal {O}}_X^S, {\mathcal {M}}) = 0 = \mathop {{\rm Ext}}\nolimits ^2_X({\mathcal {O}}_X^S, {\mathcal {M}})$, we have $\mathop {{\rm Hom}}\nolimits _X({\mathcal {M}}, Lg^*Rg_*{\mathcal {M}}[-1]) \simeq \mathop {{\rm Hom}}\nolimits _X({\mathcal {M}}, {\mathcal {O}}_X^S)$ and $\mathop {{\rm Hom}}\nolimits _X(Lg^*Rg_* {\mathcal {M}}[-1], {\mathcal {M}}) \simeq \mathop {{\rm Hom}}\nolimits _X({\mathcal {O}}_X^S, {\mathcal {M}})$. Hence, a morphism ${\mathcal {M}} \to {\mathcal {M}}$ factors via $Lg^*Rg_* {\mathcal {M}}[-1]$ if and only if it factors via ${\mathcal {O}}_X^S$.

Therefore, the image of $\psi$ is contained in the space of morphisms ${\mathcal {M}} \to {\mathcal {M}}$ that factor via ${\mathcal {O}}_X^S$. Conversely, for a projective generator ${\mathcal {P}}$ of $\mathscr {A}_f$,

\begin{align*} R\mathop{{\rm Hom}}\nolimits_{{\mathcal{D}}^b(\mathscr{A}_f)}(\Psi^!{\mathcal{M}}, \Psi^!{\mathcal{O}}_X) &\simeq R\mathop{{\rm Hom}}\nolimits_{{\mathcal{D}}^b(\mathscr{A}_f)}({\mathcal{P}}, \Psi^!{\mathcal{O}}_X[2]) \simeq R\mathop{{\rm Hom}}\nolimits_X(\Psi{\mathcal{P}}, {\mathcal{O}}_X[2]) \\ &\simeq R\mathop{{\rm Hom}}\nolimits_X(\Psi{\mathcal{P}}, f^! {\mathcal{O}}_Y[2]) \simeq R\mathop{{\rm Hom}}\nolimits_Y(Rf_* \Psi {\mathcal{P}}, {\mathcal{O}}_Y[2]) = 0, \end{align*}

which implies that the space of morphisms ${\mathcal {M}} \to {\mathcal {M}}$ that factor via direct sums of copies of ${\mathcal {O}}_X$ is annihilated by $\Psi ^!$, i.e. it is contained in the kernel of $\varphi$.