A CATEGORICAL QUANTUM TOROIDAL ACTION ON THE HILBERT SCHEMES

Abstract

We categorify the commutation of Nakajima’s Heisenberg operators $P_{\pm 1}$ andtheir infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl_1})$ acting on the Grothendieck groups of Hilbert schemes from [10, 24, 26, 32]. By combining our result with [26], one obtains a geometric categorical $U_{q_1,q_2}(\ddot {gl_1})$ action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semidivisorial log terminal singularities.

1 Introduction

1.1 The description of our result

The quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl}_1)$ (Definition 1.3) is an affinisation of the quantum Heisenberg algebra which has been realised in several contexts:

• • the elliptic Hall algebra in [2, 30],

• • the double shuffle algebra in [8, 9, 23],

• • the trace of the deformed Khovanov Heisenberg category in [4](when $q_1=q_2$ ).

Given a smooth quasiprojective surface S over $k=\mathbb {C}$ , let

$$ \begin{align*}\mathcal{M}=\bigsqcup_{n=0}^{\infty} S^{[n]}\end{align*} $$

be the Hilbert schemes of points on S. Schiffmann-Vasserot [32], Feigin-Tsymbaliuk [10] and Neguţ [26] constructed the $U_{q_1,q_2}(\ddot {gl}_1)$ action on the Grothendieck group of $\mathcal {M}$ . It generalises the action of

• • the Heisenberg algebra (Nakajima [22] and Grojnowski [11])

• • the W algebra (Li-Qin-Wang [21])

on the cohomology of Hilbert schemes.

The main result of this paper is a weak categorification of the above quantum toroidal algebra action. First let us state our results precisely. Given a nonnegative integer n, let $S^{[n,n+1]}$ be the nested Hilbert scheme

$$ \begin{align*}S^{[n,n+1]}:=\{(\mathcal{I}_n,\mathcal{I}_{n+1},x)\in S^{[n]}\times S^{[n+1]}\times S|\mathcal{I}_{n+1}\subset \mathcal{I}_n, \mathcal{I}_n/\mathcal{I}_{n+1}=k_x\},\end{align*} $$

which is a closed subscheme of $S^{[n]}\times S^{[n+1]}\times S$ . There is a tautological line bundle ${\mathcal {L}}$ on $S^{[n,n+1]}$ , such that its fibre at each closed point is $\mathcal {I}_{n}/\mathcal {I}_{n+1}$ . We abuse the notation to denote $\mathcal {I}$ as the universal ideal sheaf on ${\mathcal {M}}{\times }S$ . We will write $S_1,S_2$ for two copies of S, in order to emphasise the factors of $S\times S$ . Let ${\Delta }:{\mathcal {M}}{\times }S{\to }{\mathcal {M}}{\times }{\mathcal {M}}{\times }S_{1}{\times }S_{2}$ be the diagonal embedding and $\iota :{\mathcal {M}}{\times }{\mathcal {M}}{\times }S_{1}{\times }S_{2}{\to }{\mathcal {M}}{\times }{\mathcal {M}}{\times }S_{2}{\times }S_{1}$ be the involution map which changes the order of two copies of S. We denote $D^{b}(X)$ to be the (bounded) derived categories of coherent sheaves on a given scheme X. We prove that

Theorem 1.1 (Theorem 5.1, see Section 1.8 for the notations).

Consider the Fourier-Mukai kernels $e_k,f_k:D^b(\mathcal {M})\to D^b(\mathcal {M}\times S)$ induced from:

$$ \begin{align*} e_{k}:=\mathcal{L}^{k}\mathcal{O}_{S^{[n,n+1]}}\in D^b(S^{[n]}\times S^{[n+1]}\times S) \\ f_k:=\mathcal{L}^{k-1}\mathcal{O}_{S^{[n,n+1]}}[1]\in D^b(S^{[n+1]}\times S^{[n]}\times S). \end{align*} $$

1. (1) For every two integers m and r, there exists natural transformations

   (1.1) $$ \begin{align} \begin{cases} f_re_{m-r}\to \iota_{*}e_{m-r}f_r & \text{ if } m>0 \\ \iota_{*}e_{m-r}f_r\to f_re_{m-r} & \text{ if } m<0 \\ f_re_{-r}=\iota_{*}e_rf_{-r}\oplus \mathcal{O}_{\Delta}[1] \end{cases} ,\end{align} $$

   where $\Delta $ is the diagonal of $\mathcal {M}\times \mathcal {M}\times S \times S$ .

2. (2) When $m\neq 0$ , the cone of the natural transformations in (1.1) has a filtration with associated graded object

   $$ \begin{align*} \begin{cases} \bigoplus\limits_{k=0}^{m-1}R\Delta_*(h_{m,k}^{+}) & \text{ if } m>0 \\ \bigoplus\limits_{k=m+1}^{0}R\Delta_*(h_{m,k}^{-}) & \text{ if } m<0 \end{cases} ,\end{align*} $$

   where $h_{m,k}^{+},h_{m,k}^{-}\in D^{b}(\mathcal {M}\times S)$ are complexes of wedge and symmetric product of universal sheaves on $\mathcal {M}\times S$ .

It is natural to expect that Theorem 1.1 should be compatible with the computation in [24], which we show in Theorem 3.2.

The nontriviality of the extension is a feature of the derived category statement, which is not visible at the level of Grothendieck groups. Proposition 6.3 provides a precise extension formula.

1.2 The weak categorification of an algebra action

Categorification can take place in a very general setting. Roughly speaking, it lifts a certain quantity to a chain complex whose Betti number is the quantity. Here we follow the notation of Savage [29] for the naive, weak and strong categorifications of algebra and representations.

Let A be a commutative ring and $B=(\{b_{i}\}_{i,\in I},\{c_{j}\}_{j\in J})$ be a unitial associative A-algebra, where $\{b_{i}\}_{i\in I}$ is the generating set and $\{c_{j}\}_{j\in J}$ is a set of relations which generate all the relations in B. Let M be a B-module. The action of each $b_{i}$ defines an endomorphism $b_{i}^{M}$ of M and each $c_{j}$ defines a relation in the endomorphism ring of M which we denote as $c_{j}^{M}$ . For a triangulated category $\mathcal {M}$ , we denote $K_{0}({\mathcal {M}})$ as the Grothendieck group of ${\mathcal {M}}$ . For any $F\in {\mathcal {M}}$ , we denote $[F]\in K_{0}({\mathcal {M}})$ as the class generated by F.

Definition 1.2 (Weak categorification).

A weak categorification of $(B,\{b_{i}\},\{c_{i}\},M)$ is a quadruple $({\mathcal {M}},\phi ,\{F_{i}\}_{i\in I},\{E_{i}\}_{j\in J})$ , where

1. (1) ${\mathcal {M}}$ is a triangulated category with an isomorphism $\phi :K_{0}({\mathcal {M}})\to M$ ;

2. (2) for each $i\in I$ , $F_{i}:{\mathcal {M}}\to {\mathcal {M}}$ is an triangulated endofunctor of ${\mathcal {M}}$ , such that $[F_{i}]=b_{i}^{M}$ under the isomorphism $\phi $ ;

3. (3) for each $j\in J$ , $E_{j}=\{E_{j}^{1}\xrightarrow {e_{j}^{1}}E_{j}^{2}\xrightarrow {e_{j}^{2}}E_{j}^{3}\}$ , where $E_{j}^{1},E_{j}^{2},E_{j}^{3}$ are endofunctors of ${\mathcal {M}}$ which are generated by $F_{i}$ , $e_{j}^{1}$ and $e_{j}^{2}$ are natural transformations, such that for each element $K\in {\mathcal {M}}$ , $E_{j}^{1}(K)\xrightarrow {e_{j}^{1}(K)}E_{j}^{2}(K)\xrightarrow {e_{j}^{3}(K)}E_{j}^{3}(K)$ is an exact triangle in ${\mathcal {M}}$ and the relation $c_{j}^{M}=\{[E_{j}^{1}]-[E_{j}^{2}]+[E_{j}^{3}]=0\}$ under the isomorphism $\phi $ .

Now we recall an integral version of the quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl}_1)$ .

Definition 1.3 ([26]).

Given two formal parameters $q_1$ and $q_2$ , let $q=q_1q_2$ . Let $\mathbb {K}=\mathbb {Z}[q_1+q_{2},q,q^{-1}]$ . The quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl_1})$ is the $\mathbb {K}$ -algebra with generators:

$$ \begin{align*} \{E_{k},F_{k},H_{l}^{\pm}\}_{k\in \mathbb{Z},l\in \mathbb{N}} \end{align*} $$

modulo the following relations:

(1.2) $$ \begin{align} (z-wq_1)(z-wq_2)(z-\frac{w}{q})E(z)&E(w)= \\ &\quad =(z-\frac{w}{q_1})(z-\frac{w}{q_2})(z-wq)E(w)E(z) \notag \end{align} $$

(1.3) $$ \begin{align} (z-wq_1)(z-wq_2)(z-\frac{w}{q})E(z)&H^{\pm}(w)= \\ & \ \ =(z-\frac{w}{q_1})(z-\frac{w}{q_2})(z-wq)H^{\pm}(w)E(z) \notag \end{align} $$

(1.4) $$ \begin{align} [[E_{k+1},E_{k-1}],E_k]=0 \quad \forall k\in \mathbb{Z} \end{align} $$

together with the opposite relations for $F(z)$ instead of $E(z)$ , as well as:

(1.5) $$ \begin{align} [E(z),F(w)]=\delta(\frac{z}{w})(1-q_1)(1-q_2)(\frac{H^+(z)-H^{-}(w)}{1-q}) ,\end{align} $$

where

(1.6) $$ \begin{align} E(z)=\sum_{k\in \mathbb{Z}}\frac{E_k}{z^k}, \quad F(z)=\sum_{k\in \mathbb{Z}}\frac{F_k}{z^k}, \quad H^{\pm}(z)=\sum_{l\in \mathbb{N}\cup \{0\}}\frac{H_l^{\pm}}{z^{\pm l}} ,\end{align} $$

where

$$ \begin{align*} \delta(z)=\sum_{n\in \mathbb{Z}}z^n\in \mathbb{Q}\{\{z\}\}. \end{align*} $$

We will set $H_0^+=q$ and $H_0^{-}=1$ .

The weak categorification of relations (1.2), (1.3), (1.4) was obtained in the main theorem of [26]. Hence, we still need to categorify (1.5) to obtain a weak categorification of the quantum toroidal algebra action, which is the purpose of Theorem 1.1.

1.3 Outline of the proof

Let us first review the categorification of the commutation of $e_k$ and $e_l$ for different k and l in [26]. Consider the moduli spaces $\mathfrak {Z}_2,\mathfrak {Z}_{2}'$ which parameterise diagrams

(1.7)

(1.8)

respectively, of ideal sheaves, where each successive inclusion is colength $1$ and supported at the point indicated on the diagrams. Then $e_ke_l$ and $e_le_k$ are the derived pushfoward of line bundles on $\mathfrak {Z}_2$ and $\mathfrak {Z}_2'$ to $S^{[n-1]}\times S^{[n+1]}\times S\times S$ , respectively. In order to compare $e_ke_l$ and $e_le_k$ , [26] introduced the quadruple moduli space $\mathfrak {Y}$ which parameterises the diagram

(1.9)

of ideal sheaves, where each successive inclusion is colength $1$ and supported at the point indicated on the diagrams. $\mathfrak {Y}$ is smooth and induces resolutions of $\mathfrak {Z}_{2}$ and $\mathfrak {Z}_{2}'$ . Proposition 2.30 of [26] proved that $\mathfrak {Z}_{2}$ and $\mathfrak {Z}_{2}'$ are rational singularities, based on the fact that any fibre of the resolution has dimension $\leq 1$ . Thus, $e_ke_l$ and $e_le_k$ could be compared through line bundles on $\mathfrak {Y}$ .

Now in order to compare $f_re_{m-r}$ and $e_{m-r}f_r$ , we introduce the triple moduli spaces $\mathfrak {Z}_+,\mathfrak {Z}_{-}$ which parameterise diagrams

(1.10)

(1.11)

Then $f_re_{m-r}$ and $e_{m-r}f_r$ are the derived pushforward of line bundles on $\mathfrak {Z}_+,\mathfrak {Z}_{-}$ , respectively. The quadruple moduli space $\mathfrak {Y}$ still induces resolutions of $\mathfrak {Z}_{-}$ , but $\mathfrak {Z}_{+}$ have two irreducible components. One irreducible component is $S^{[n,n+1]}$ , denoted by $W_1$ , and the other irreducible component is denoted by $W_0$ in Section 6. $\mathfrak {Y}$ induces a resolution of $W_0$ .

In order to compare $f_re_{m-r}$ and $e_{m-r}f_r$ through line bundles on $\mathfrak {Y}$ , one must prove that $\mathfrak {Z}_{-}$ and $W_0$ are rational singularities. The approach in [26] did not work here, as the fibre could have pretty large dimensions. Instead, we study the singularity structure of $\mathfrak {Z}_{+}$ and $\mathfrak {Z}_-$ through the viewpoint of the minimal model program (MMP) [17, 18]. We prove that

Proposition 1.4 (Propositions 4.6 and 4.13).

The pair $(\mathfrak {Z}_+,0)$ is semi-dlt. $\mathcal {Z}_{-}$ and $W_0$ are canonical singularities.

We prove Proposition 1.4 by explicitly computing the discrepancy (see Section 6 for the definitions of semi-dlt, canonical singularities and the discrepancy). Canonical singularities are always rational singularities by Theorem B.7.

Remark 1.5. One should notice the elliptic Hall algebra of [2] contains more relations than Definition 1.3, which we would investigate in the future. The main obstacle of generalising our result to the elliptic Hall algebra is that for the action of other operators, the corresponding nested moduli space is not Cohen-Macaulay, and, hence, the enhancement in derived algebraic geometry has to be considered. It is also the obstacle of generalising our result to the quantum toroidal algebra action on the Grothendieck group of higher rank stable sheaves [26], as $\mathfrak {Z}_+$ is no longer equidimensional in this situation.

1.4 Categorical Heisenberg actions

Khovanov [16] defined the Heisenberg category through graphical calculus. Cautis-Licata [5] constructed a categorical Heisenberg action on the derived category of Hilbert schemes of points of the minimal resolution of the type ADE singularities. Krug [20] constructed the weak categorical Heisenberg action on the derived category of Hilbert schemes of points on smooth surfaces. Our categorification is different from those above, as it is given in terms of explicit correspondences and independent of the derived McKay correspondence.

Although the higher Nakajima operators were categorified by the objects $e_{(0,\ldots ,0)}$ of [26], the relations between them (as well as the morphisms between them in Khovanov’s Heisenberg category) are still unclear to us.

1.5 Double categorified Hall algebra

The study of Cohomological Hall algebra was initiated by Kontsevich-Soibelman [19] and Schiffmann-Vasserot [31]. Kapranov-Vasserot [15] and the author [37] constructed the K-theoretic Hall algebra on surfaces, which was categorified by Porta-Sala [27]. It also categorified the positive half of $U_{q}(\ddot {gl}_1)$ when $S=\mathbb {A}^2$ . The relation between the categorified Hall algebra of minimal resolution of type A singularities and quivers was studied by Diaconescu-Porta-Sala [6]. On the other hand, the Drinfeld double of the categorified Hall algebra is still mysterious. As an attempt to understand the action of the ‘double Categorified Hall algebra’, it is natural to expect that our approach could be generalised to categorifications in other settings, like those of Toda [36] and Rapcak-Soibelman-Yang-Zhao [28].

1.6 Other related work

Recently, Addington-Takahashi [35] studied certain sequences of moduli spaces of sheaves on $K3$ surfaces and showed that these sequences can be given the structure of a geometric categorical $\mathfrak {sl}_2$ action in the sense of [3]. It would be interesting to explore the interactions between their action and ours.

Another related work is Jiang-Leung’s projectivisation formula [14]. Through this formula, they obtained a semiorthogonal decomposition of the derived category of the nested Hilbert schemes.

1.7 The organisation of the paper

The proof of the main theorem is in Section 5 and the extension formula is in Section 6. The other sections are organised as follows:

• Section 2: we review the action of $U_{q_1,q_2}(\ddot {gl}_1)$ on the Grothendieck group of Hilbert schemes [10, 24, 26, 32];

• Section 3: we define $h_{m,k}^{+}\in D^b(\mathcal {M}\times S)$ and prove the third part of Theorem 1.1;

• Section 4: we study the singularity structures of $\mathfrak {Z}_{-}$ and $\mathfrak {Z}_{+}$ through the singularity theory of the minimal model program.

1.8 Notations

In this paper, we will always work over $k=\mathbb {C}$ .

1.8.1 Derived categories and the Grothendieck groups

For any scheme X, we denote $D_{qcoh}(X)$ as the derived category of quasicoherent sheaves on X. We denote $D^{u}(X)$ as the full subcategory of $D_{qcoh}(X)$ which consists of elements, such that all the cohomologies are coherent sheaves on X. We denote $D^{b}(X)$ as the full subcategory of $D^{u}(X)$ , such that the cohomologies are bounded. We denote $K(X):=K_{0}(D^{b}(X))$ .

1.8.2 Fourier-Mukai transforms associated to a surface

We will write $S_1,S_2$ for two copies of S, in order to emphasise the factors of $S\times S$ and write $\mathcal {M}_1$ , $\mathcal {M}_2$ and $\mathcal {M}_3$ for three copies of $\mathcal {M}$ , in order to emphasise the factors of $\mathcal {M}\times \mathcal {M}\times \mathcal {M}$ . We define elements P in $D^{b}(\mathcal {M}_{1}\times \mathcal {M}_{2}\times S)$ to be the Fourier-Mukai kernels associated to S. Given $P\in D^{b}(\mathcal {M}_{1}\times \mathcal {M}_{2}\times S_1)$ and $Q\in D^{b}(\mathcal {M}_{2}\times \mathcal {M}_{3}\times S_2)$ , we define the composition $QP\in D^{b}(\mathcal {M}_{1}\times \mathcal {M}_{3}\times S_1\times S_2)$ by

$$ \begin{align*}QP:=\mathbf{R}\pi_{13_{*}}(\mathbf{L}\pi_{12}^*P\otimes \mathbf{L}\pi_{23}^*Q),\end{align*} $$

where $\pi _{12},\pi _{23}$ and $\pi _{13}$ are the projections from $\mathcal {M}_1\times \mathcal {M}_2\times \mathcal {M}_3\times S_1\times S_2$ to $\mathcal {M}_1\times \mathcal {M}_2\times S_1$ , $\mathcal {M}_2\times \mathcal {M}_3\times S_2$ and $\mathcal {M}_1\times \mathcal {M}_3\times S_1\times S_2$ , respectively.

1.8.3 Complexes

In this paper, we adapt the cohomological degree for complexes, that is the degree of a complex is always increasing. For any complex

$$ \begin{align*}\{\cdots \to C_{-1}\to C_0\to 0\},\end{align*} $$

we will assume that $C_0$ has cohomological degree $0$ unless explicitly pointing out the cohomological degree. Given a two term complex of locally free sheaves $U:=\{W\xrightarrow {s}V\}$ , we denote the symmetric product and the wedge product complexes:

$$ \begin{align*} S^{k}(U):=\{\wedge^{k}W\cdots \to \cdots \to W\otimes S^{k-1}V\to S^k(V) \}\\ \wedge^{k}(U):=\{S^{k}W\to \cdots \to \wedge^{k-1}(V)\otimes W\to \wedge^k(V)\} \end{align*} $$

and $S^{k}(U)=\wedge ^{k}(U)=0$ when $k<0$ . At the level of Grothendieck groups, we have

$$ \begin{align*}[\wedge^k(U)]:=\sum_{i=0}^k(-1)^i[S^iW][\wedge^{k-i}V] \quad [S^k(U)]:=\sum_{i=0}^k(-1)^i[\wedge^iW][S^{k-i}V].\end{align*} $$

We define $det(U):=\frac {det(V)}{det(W)}$ and $U^{\vee }$ as the two term complex

$$ \begin{align*}\{V^{\vee}\xrightarrow{\mathfrak{u}^{\vee}}W^{\vee}\}.\end{align*} $$

Given complexes $\{C_i| i\in \mathbb {Z}\}$ with morphisms $d_i:C_i\to C_{i+1}$ , such that $d_i\circ d_{i+1}=0$ , we will write

$$ \begin{align*}\{\cdots \to C_{i+1}\to C_i\to \cdots \}\end{align*} $$

for the total complex of the double complex $C_{\bullet }$ . Given a complex $C_{\bullet }$ and an integer k, we denote the complex $C_{\bullet }[k]$ , such that the degree n term is $C_{n+k}$ .

2 The quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl}_1)$ and the K-theory of Hilbert scheme of points on surfaces

In this section, we will review the action of $U_{q_1,q_2}(\ddot {gl}_1)$ on the K-theory on Hilbert scheme of points on surfaces from [10, 24, 26, 32]. The main theorem will be formulated in Theorem 2.6.

2.1 Hilbert and nested Hilbert schemes

Given an integer $n>0$ and a smooth quasiprojective surface S over $k=\mathbb {C}$ , let

$$ \begin{align*}S^{[n]}:=\{\mathcal{I}_n\subset \mathcal{O}|\mathcal{O}/\mathcal{I}_n \text{ is dimension } 0 \text{ and length } n\}\end{align*} $$

be the Hilbert scheme of n points on S. There is a universal ideal sheaf on $S^{[n]}\times S$ , which we still denote as $\mathcal {I}_n$ . Let $\mathcal {Z}_n\subset S^{[n]}\times S$ be the closed scheme of $S^{[n]}\times S$ with the ideal sheaf $\mathcal {I}_{n}$ . We define the Hilbert schemes of points on S as

$$ \begin{align*}\mathcal{M}:=\bigsqcup_{n=0}^{\infty} S^{[n]}.\end{align*} $$

Proposition 2.1 (Proposition 2.11 of [26]).

There exists a resolution of $\mathcal {I}_{n}$ by

(2.1) $$ \begin{align} 0\to W_n \xrightarrow{s} V_n \to \mathcal{I}_{n}\to 0 ,\end{align} $$

where $W_n$ and $V_n$ are locally free coherent sheaves with the same determinant. Let $w_n$ and $v_n$ be the rank of $W_n$ and $V_n$ , respectively. Then $v_n-w_n=1$ .

Definition 2.2. The nested Hilbert scheme $S^{[n,n+1]}$ is defined to be

$$ \begin{align*} S^{[n,n+1]}:=\{(\mathcal{I}_n,\mathcal{I}_{n+1},x)\in S^{[n]}\times S^{[n+1]}\times S|\mathcal{I}_{n+1}\subset \mathcal{I}_n, \mathcal{I}_n/\mathcal{I}_{n+1}=k_x\} \end{align*} $$

with natural projection maps

(2.2)

(2.3)

and let

$$ \begin{align*} p_n:=(p_{n}^+,\pi_n): S^{[n,n+1]}\to S^{[n]}\times S. \end{align*} $$

We abuse the notation to denote ${\mathcal {I}}_{n}$ and ${\mathcal {I}}_{n+1}$ as the universal sheaf on $S^{[n,n+1]}{\times }S$ . Then ${\mathcal {I}}_{n+1}\subset {\mathcal {I}}_{n}$ and the pushforward of ${\mathcal {I}}_{n}/{\mathcal {I}}_{n+1}$ to $S^{[n,n+1]}$ is a line bundle, which we denote as $\mathcal {L}$ . The fibre of $\mathcal {L}$ at each closed point $({\mathcal {I}}_{n+1}\subset {\mathcal {I}}_{n})$ is $\mathcal {I}_{n}/\mathcal {I}_{n+1}$ .

2.2 The quantum toroidal algebra action on the K-theory of Hilbert schemes

Definition 2.3 (Definitions 4.10 and 4.11 of [26]).

Let $\Delta _S:S\to S\times S$ be the diagonal embedding. For any group homomorphisms $x,y:K(\mathcal {M})\to K(\mathcal {M}\times S)$ , we define:

$$ \begin{align*}xy|_{\Delta_S}=\{K(\mathcal{M})\xrightarrow{y} K(\mathcal{M}\times S)\xrightarrow{x\times Id_{S}}K(\mathcal{M}\times S\times S)\xrightarrow{Id_{\mathcal{M}}\times \Delta_S^*}K(\mathcal{M}\times S)\}\end{align*} $$

$$ \begin{align*} &[x,y]=\{K(\mathcal{M})\xrightarrow{y} K(\mathcal{M}\times S_2)\xrightarrow{x\times Id_{S_2}}K(\mathcal{M}\times S_1\times S_2)\} \\ &\qquad\qquad\qquad\qquad\qquad -\{K(\mathcal{M})\xrightarrow{x} K(\mathcal{M}\times S_1)\xrightarrow{y\times Id_{S_1}}K(\mathcal{M}\times S_1\times S_2)\} .\end{align*} $$

We define

$$ \begin{align*}[x,y]_{red}=z\end{align*} $$

for a group homomorphism $z:K(\mathcal {M})\to K(\mathcal {M\times S})$ if

$$ \begin{align*} [x,y]=\Delta_{S*}(z). \end{align*} $$

The definition is unambiguous, since $\Delta _{S*}:K(S)\to K(S\times S)$ is injective, and so z is unique.

Definition 2.4. Let $\omega _{S}$ be the canonical line bundle of S. We define $h_0^{+}:=[\omega _S]$ , $h_{0}^{-}:=1$ , and when $m>0$

(2.4) $$ \begin{align} h_{m}^{+}:=(1-\omega_S) \sum_{j=0}^{m-1}[\omega_S^{-j}]\sum_{i=0}^{j}(-1)^{i}[S^{m-i}\mathcal{I}_n][\wedge^{i}\mathcal{I}_n] \end{align} $$

(2.5) $$ \begin{align} h_m^{-}:=(1-\omega_S)\sum_{j=0}^{m-1}(-1)^{j}[\omega_S^{j}]\sum_{i=0}^j(-1)^i[\wedge^{m-i}\mathcal{I}_n^{\vee}][S^i\mathcal{I}_n^{\vee}] \end{align} $$

as elements in $K(S^{[n]}\times S).$ Here, we abuse the notation to denote

$$ \begin{align*} \mathcal{I}_n:=\{W_n\xrightarrow{s}V_n\} \end{align*} $$

in the short exact sequence (2.1).

Remark 2.5. Definition 2.4 is equivalent to the definition of $h_{m}^{\pm }$ in [24]. We will prove it in Appendix A.

Theorem 2.6 (Theorem 1.2 of [24]).

Let $T^*S$ be the cotangent bundle of S and $\omega _S$ be the canonical bundle of S. The morphism:

$$ \begin{align*} q_1+q_2\to [T^*S] \quad q=q_1q_2\to [\omega_S] \end{align*} $$

induces a homomorphism:

$$ \begin{align*}\mathbb{K}\to K(S).\end{align*} $$

We regard $S^{[n,n+1]}$ as a closed subscheme of $S^{[n]}{\times }S^{[n+1]}{\times }S$ and $S^{[n]}{\times }S$ as a closed subscheme of $S^{[n]}{\times }S^{[n]}{\times }S$ through the diagonal embedding and consider the following element:

$$ \begin{align*} \tilde{e}_i:=[\mathcal{L}^i\mathcal{O}_{S^{[n,n+1]}}]\in K(S^{[n]}\times S^{[n+1]}\times S),\\ \tilde{f}_i:=-[\mathcal{L}^{i-1}\mathcal{O}_{S^{[n,n+1]}}]\in K(S^{[n+1]}\times S^{[n]}\times S), \\ \tilde{h}_{i}^{\pm}:=[h_{i}^{\pm}\mathcal{O}_{S^{[n]}\times S}]\in K(S^{[n]}\times S^{[n]}\times S). \end{align*} $$

All the elements $\tilde {e}_{i}$ , $\tilde {f}_{i}$ and $h_{i}^{\pm }$ could be regarded as operators $K(\mathcal {M})\to K(\mathcal {M}\times S)$ through the K-theoretic correspondences. Then there exists a unique $\mathbb {K}$ -homomorphism

$$ \begin{align*}\Phi:U_{q_1,q_2}(\ddot{gl}_1)\to Hom(K_{\mathcal{M}},K_{\mathcal{M}\times S}),\end{align*} $$

such that

1. (1)

   $$ \begin{align*}\Phi(E_i)=\tilde{e}_i, \quad \Phi(F_i)=\tilde{f}_i, \quad \Phi(H_i^{\pm})=\tilde{h}_i^{\pm}\end{align*} $$

2. (2) For all $x,y\in U_{q_1,q_2}(\ddot {gl}_1)$ , we have

   $$ \begin{align*} \Phi(xy)=\Phi(x)\Phi(y)|_{\Delta_S} \\ [\Phi(x),\Phi(y)]|_{red}=\Phi(\frac{[x,y]}{(1-q_1)(1-q_2)}). \end{align*} $$

   The right-hand side is well defined due to the fact that all commutators in $U_{q_1,q_2}(\ddot {gl}_1)$ are multiples of $(1-q_1)(1-q_2)$ (see Theorem 2.4 of [25]).

3 Nested Hilbert schemes and $h_{m,k}^{\pm }$

In this section, we consider the nested Hilbert scheme $S^{[n-1,n,n+1]}$ by the Cartesian diagram:

(3.1)

which consists of

$$ \begin{align*}\{(\mathcal{I}_{n-1},\mathcal{I}_n,\mathcal{I}_{n+1},x)\in S^{[n-1]}\times S^{[n]}\times S^{[n+1]}\times S|\mathcal{I}_{n-1}/\mathcal{I}_{n}=k_x,\mathcal{I}_{n}/\mathcal{I}_{n+1}=k_x\}.\end{align*} $$

Like the definition of the line bundle ${\mathcal {L}}$ in Definition 2.2, we can also define two line bundles $\mathcal {L}_1,\mathcal {L}_{2}$ on $S^{[n-1,n,n+1]}$ whose fibres are $\mathcal {I}_{n}/\mathcal {I}_{n+1},\mathcal {I}_{n-1}/\mathcal {I}_n$ , respectively. We denote

$$ \begin{align*} q_n:S^{[n-1,n,n+1]}\to S^{[n,n+1]} \end{align*} $$

as the projection morphism.

Example 3.1. Let $\Delta _{S}:S\to S\times S$ be the diagonal embedding and $\mathcal {I}_{\Delta _S}$ be the ideal sheaf of the diagonal. Then

$$ \begin{align*} S^{[1,2]}=Bl_{\Delta_S}(S\times S)=\mathbb{P}_{S\times S}(\mathcal{I}_{\Delta_{S}}) \qquad S^{[2]}=Bl_{\Delta_{S}}(S\times S)/\mathbb{Z}_2 , \end{align*} $$

where the $\mathbb {Z}_2$ action on $Bl_{\Delta _{S}}(S{\times }S)$ is induced by the $\mathbb {Z}_2$ action

$$ \begin{align*} i:S\times S\to S\times S \quad i(x,y)=(y,x). \end{align*} $$

By [33], the projection morphism

$$ \begin{align*} (p_{2}^-,\pi_2):S^{[1,2]}\to S^{[2]}\times S\quad (\mathcal{I}_1,\mathcal{I}_2,x)\to (\mathcal{I}_2,x) \end{align*} $$

is a closed embedding with image $\mathcal {Z}_2$ . By (3.1), $S^{[1,2,3]}$ is the preimage of $\mathcal {Z}_2$ .

For two integers $k,m$ , such that $m>k\geq 0$ , we define $h_{m,k}^{+}\in D^{b}(S^{[n]}\times S)$ by

(3.2) $$ \begin{align} h_{m,k}^+:= \begin{cases} \mathbf{R}(p_n\circ q_n)_*(\mathcal{L}_1^{m-1-k}\mathcal{L}_2^k)[1] & k>0 \\ \mathbf{R}p_{n*}(\mathcal{L}^{m})[2] & k=0 \end{cases} \end{align} $$

and if $m<k \leq 0$ , we define $h_{m,k}^-\in D^{b}(S^{[n]}\times S)$ by

(3.3) $$ \begin{align} h_{m,k}^{-}:= \begin{cases} \mathbf{R}(p_n\circ q_n)_*(\mathcal{L}_1^{m-1-k}\mathcal{L}_2^{k})[1] & k<0 \\ \mathbf{R}p_{n*}(\mathcal{L}^{-m-1})[1] & k=0. \end{cases} \end{align} $$

The purpose of this section is to prove that

Theorem 3.2. At the level of Grothendieck groups,

$$ \begin{align*} [h_{m,k}^+]=[\omega_S^{-k}]\sum_{i=0}^{k}(-1)^{i}[S^{m-i}\mathcal{I}_n][\wedge^{i}\mathcal{I}_n] \\ [h_{m,k}^-]=(-1)^{k}[\omega_S^{k}]\sum_{i=0}^k(-1)^i[\wedge^{m-i}\mathcal{I}_n^{\vee}][S^i\mathcal{I}_n^{\vee}] \end{align*} $$

and at the level of Grothendieck groups

(3.4) $$ \begin{align} \begin{cases} (1-[\omega_S])\sum\limits_{k=0}^{m-1}[h_{m,k}^+]=h_m^{+} & \qquad m>0 \\ (1-[\omega_S])\sum\limits_{k=m+1}^{0}[h_{m,k}^-]=h_{-m}^{-} & \qquad m<0. \end{cases} \end{align} $$

3.1 Projectivisation and A categorical projection lemma

Definition 3.3. Let

$$ \begin{align*} U:=\{W\xrightarrow{s} V\} \end{align*} $$

be a two term complex of locally free sheaves over a scheme X, such that W has rank w and V has rank v. Let $Z\subset \mathbb {P}_X(V)$ be the closed subscheme, such that $\mathcal {O}_Z$ is the cokernel of the composition of morphisms

$$ \begin{align*} \rho^*W\otimes \mathcal{O}_{\mathbb{P}_X(V)}(-1)\xrightarrow{\rho^*(s)}\rho^*V\otimes \mathcal{O}_{\mathbb{P}_X(V)}(-1)\xrightarrow{taut} \mathcal{O}_{\mathbb{P}_X(V)}, \end{align*} $$

where $\rho :\mathbb {P}_X(V)\to X$ is the projection morphism. We define Z to be the projectivisation of U over X, denoted by

$$ \begin{align*} Z=\mathbb{P}_X(U) \end{align*} $$

if $\mathcal {O}_Z$ is resolved by the Koszul complex:

$$ \begin{align*}0 \to \wedge^w\rho^*(W)\otimes \mathcal{O}_{\mathbb{P}_X(V)}(-w) \to \cdots \to \rho^*W\otimes \mathcal{O}_{\mathbb{P}_X(V)}(-1)\to \mathcal{O}_{\mathbb{P}_X(V)}\to \mathcal{O}_Z\to 0.\end{align*} $$

When Z is a projectivisation of U over X, we have a categorical projection lemma for $\mathbf {R}\rho _{*}(\mathcal {O}_Z(k))$ :

Lemma 3.4 (Categorical projection lemma).

If Z is the projectivisation of U over X in Definition 3.3, then the tensor contraction

(3.5) $$ \begin{align} det(U)^{-1}\wedge^{w-v-k}W^{\vee}=\wedge^v(V^{\vee})\wedge^{k+v}(W)\to \wedge^k(W) \end{align} $$

induces a morphism of complexes

(3.6) $$ \begin{align} det(U)^{-1}\wedge^{w-v-k}(U^{\vee})[k] \to S^k(U) \end{align} $$

and $\mathbf {R}\rho _*(\mathcal {O}_Z(k))$ is quasi-isomorphic to its cone.

Proof. $\mathcal {O}_Z(k)$ is quasi-isomorphic to the complex

$$ \begin{align*}\{\cdots \to \wedge^j\rho^*W\otimes \mathcal{O}_{\mathbb{P}_X(V)}(-j+k) \to \cdots \to \rho^*W\otimes \mathcal{O}_{\mathbb{P}_X(V)}(k-1)\to \mathcal{O}_{\mathbb{P}_X(V)}(k)\to 0\}.\end{align*} $$

Consider the following two complexes

$$ \begin{align*} F_0=\{\cdots\to \wedge^{k+v+1}\rho^*W\otimes \mathcal{O}_{\mathbb{P}_X(V)}(-v-1)\to \wedge^{k+v}\rho^*W\otimes\mathcal{O}_{\mathbb{P}_X(V)}(-v)\}[v+k-1] \\ F_1=\{\wedge^{k+v-1}\rho^*W\otimes\mathcal{O}_{\mathbb{P}_X(V)}(-v+1)\cdots\to \rho^*W\otimes \mathcal{O}_{\mathbb{P}_X(V)}(k-1)\to \mathcal{O}_{\mathbb{P}_X(V)}(k)\}. \end{align*} $$

Then the morphism $\wedge ^{k+v}\rho ^*W\otimes \mathcal {O}_{\mathbb {P}_X(V)}(-v)\to \wedge ^{k+v-1}\rho ^*W\otimes \mathcal {O}_{\mathbb {P}_X(V)}(-v+1)$ induces a morphism of $F_0\to F_1$ with the cone quasi-isomorphic to $\mathcal {O}_{Z}(k)$ .

By Exercise III.8.4 of Hartshorne [12],

(3.7) $$ \begin{align} \mathbf{R}\rho_*(\mathcal{O}_{\mathbb{P}(V)}(j))= \begin{cases} S^{j}(V) & j\geq 0 \\ 0 & -v<j<0 \\ detV^{-1}\otimes S^{-j-v}V^{\vee}[1-v] & j\leq -v, \end{cases} \end{align} $$

and thus,

$$ \begin{align*}\mathbf{R}\rho_*F_0\cong det(U)^{-1}\wedge^{w-v-k}(U^{\vee})[-k], \quad \mathbf{R}\rho_*F_1\cong S^k(U).\end{align*} $$

Hence, $\mathbf {R}\rho _*(\mathcal {O}_Z(k))$ is quasi-isomorphic to the cone

$$ \begin{align*}det(U)^{-1}\wedge^{w-v-k}(U^{\vee})[k] \to S^k(U).\\[-35pt]\end{align*} $$

3.2 Nested Hilbert schemes as projectivisation

Recall the short exact sequence (2.1):

$$ \begin{align*}0\to W_n\xrightarrow{s_n}V_n\to \mathcal{I}_n\to 0.\end{align*} $$

Nested Hilbert schemes can be realised as projectivisations, as in the following Propositions:

Proposition 3.5 (Proposition 2.2 and Lemma 3.1 of [7]).

The nested Hilbert scheme $S^{[n,n+1]}$ is the blow up of $\mathcal {Z}_n$ in $S^{[n]}\times S$ and

$$ \begin{align*}S^{[n,n+1]}\cong\mathbb{P}_{S^{[n]}\times S}(\mathcal{I}_n)\end{align*} $$

is smooth of dimension $2n+2$ . Moreover, $S^{[n,n+1]}$ is the projectivisation of

$$ \begin{align*}W_n\xrightarrow{s} V_n\end{align*} $$

over $S^{[n]}\times S$ . The tautological line bundle $\mathcal {L}$ is the restriction of $\mathcal {O}_{\mathbb {P}_{S^{[n]}\times S}(V_{n})}(1)$ to $S^{[n,n+1]}$ .

Corollary 3.6. The line bundle $\mathcal {L}$ is the exceptional divisor of $S^{[n,n+1]}$ as the blow up of $\mathcal {Z}_n$ , that is we have the short exact sequence:

(3.8) $$ \begin{align} 0\to \mathcal{L}\to \mathcal{O}_{S^{[n,n+1]}} \to p_n^{-1}\mathcal{O}_{\mathcal{Z}_n}\to 0. \end{align} $$

Proof. It is obvious from Proposition 7.13 of [12].

Let $\overline {V_{n}}$ be the kernel of the surjective morphism

$$ \begin{align*}p_{n}^*V_{n}\to \mathcal{O}_{S^{[n,n+1]}}(1)=\mathcal{L}.\end{align*} $$

Then $\overline {V_{n}}$ is also locally free. The morphism $p_n^*(W_{n})\to p_n^*(V_{n})$ factors through $\overline {V_{n}}$ and induces a morphsim

$$ \begin{align*}\overline{V_{n}}^{\vee}\otimes \omega_S\to p_n^*(W_{n}^{\vee})\otimes \omega_S.\end{align*} $$

Proposition 3.7 (Proposition 4.15 of [25]).

The scheme $S^{[n-1,n,n+1]}$ is smooth of dimension $2n+1$ . Moreover, it is the projectivisation of

$$ \begin{align*}\overline{V_{n}}^{\vee}\otimes \omega_S\to p_n^*(W_{n}^{\vee})\otimes \omega_S\end{align*} $$

over $S^{[n,n+1]}$ . For the two tautological line bundles $\mathcal {L}_1$ , $\mathcal {L}_2$ , $\mathcal {L}_1=q_n^*(\mathcal {O}_{S^{[n,n+1]}}(1))$ and $\mathcal {L}_2$ is the restriction of $\mathcal {O}_{\mathbb {P}_{S^{[n,n+1]}}(p_n^*(W_{n}^{\vee })\otimes \omega _S)}(-1)$ in $S^{[n,n+1]}$ .

3.3 The derived pushforward of line bundles on $S^{[n-1,n,n+1]}$

In this subsection, we still abuse the notation to denote

$$ \begin{align*}\mathcal{I}_n:=\{W_{n}\to V_{n}\}, \qquad \mathcal{I}_n^{\vee}:=\{V_{n}^{\vee}\to W_{n}^{\vee}\}.\end{align*} $$

Lemma 3.8. We have the following formula for the derived pushforward $\mathbf {R}p_{n*}\mathcal {L}^{j}$ :

(3.9) $$ \begin{align} \mathbf{R}p_{n*}(\mathcal{L}^j)= \begin{cases} S^j(\mathcal{I}_n) & j \geq 0 \\ \wedge^{-j-1}(\mathcal{I}_n^{\vee})[1+j] & j<0. \end{cases} \end{align} $$

Proof. By Proposition 3.5, $S^{[n,n+1]}$ is the projectivisation of

$$ \begin{align*}W_n\xrightarrow{s_n} V_n.\end{align*} $$

Thus, (3.9) follows from Lemma 3.4.

Lemma 3.9. We have the following formula for $\mathbf {R}q_{n*}\mathcal {L}_2^k$ :

(3.10) $$ \begin{align} \mathbf{R}q_{n*}(\mathcal{L}_2^k)= \begin{cases} \{\mathcal{L} \to \mathcal{O}_{S^{[n,n+1]}}\} & k=0 \\ \omega_S^{-k}\otimes \{\mathbf{L}p_n^*(\wedge^{k}\mathcal{I}_n)\mathcal{L}\to \cdots \to \mathbf{L}p_n^*(\mathcal{I}_n)\otimes \mathcal{L}^{k}\to \mathcal{L}^{k+1}\}[1-2k] & k>0 \\ \omega_S^{-k}\otimes \{\cdots \to \mathcal{L}^{-1}\mathbf{L}p_n^*(S^{-k-1}\mathcal{I}_n^{\vee})[1]\to \mathbf{L}p_n^*(S^{-k}\mathcal{I}_n^{\vee}) \} & k<0, \end{cases} \end{align} $$

where $\mathbf {L}p_n^*:D^b(S^{[n]}\times S)\to D^b(S^{[n,n+1]})$ is the derived pullback morphism.

Proof. By Proposition 3.7, $S^{[n-1,n,n+1]}$ is the projectivisation of

$$ \begin{align*}\overline{V_{n}}^{\vee}\otimes \omega_S\to p_n^*(W_{n}^{\vee})\otimes \omega_S\end{align*} $$

over $S^{[n,n+1]}$ . We notice that $p_n^*(W_{n}^{\vee })\otimes \omega _S$ and $\overline {V_{n}}^{\vee }\otimes \omega _S$ have the same rank and

$$ \begin{align*} \frac{det(p_n^*(W_{n})\otimes \omega_S)}{det(\overline{V_{n}})\otimes \omega_S}= \frac{det(p_n^*(W_{n}))}{det(\overline{V_{n}})} = \frac{det(p_n^*(V_{n}))}{det(p_n^*(V_{n}))\mathcal{L}^{-1}}=\mathcal{L}. \end{align*} $$

By Lemma 3.4, we have

• • When $k=0$ ,

  (3.11) $$ \begin{align} \mathbf{R}q_{n*}(\mathcal{O}_{S^{[n-1,n,n+1]}})=\{\mathcal{L}\to \mathcal{O}_{S^{[n,n+1]}}\}, \end{align} $$

• • When $k>0$ , we have

  $$ \begin{align*}\mathbf{R}q_{n*}\mathcal{L}_2^k=\mathcal{L}\otimes \omega_S^{-k}\otimes \{S^{k}(p_{n}^*W_{n})\to \overline{V_{n}}\otimes S^{k-1}(p_n^*W_{n})\cdots \to \wedge^{k}(\overline{V_{n}})\}[1-k].\end{align*} $$
  By the resolution of $\wedge ^k(\overline {V_{n}})$ :
  $$ \begin{align*} 0\to \wedge^{k}(\overline{V_{n}})\to \wedge^k(V_{n})\to \wedge^{k-1}(V_{n})\otimes \mathcal{L}\to \cdots \to \mathcal{L}^k\to 0, \end{align*} $$
  we have
  $$ \begin{align*} \mathbf{R}q_{n*}(\mathcal{L}_2^k)=\omega_S^{-k}\otimes \{\mathbf{L}p_n^*(\wedge^{k}\mathcal{I}_n)\mathcal{L}\to \cdots \to \mathbf{L}p_n^*(\mathcal{I}_n)\otimes \mathcal{L}^{k}\to \mathcal{L}^{k+1}\}[1-2k]. \end{align*} $$

• • When $k<0$ , we have

  $$ \begin{align*} \mathbf{R}q_{n*}(\mathcal{L}_2^k)=\omega_S^{-k}\otimes \{\cdots \to \overline{V_{n}}^{\vee}\otimes S^{-k-1}(p_n^*W_{n}^{\vee})\to S^{-k}(p_n^*W_{n}^{\vee})\} \end{align*} $$
  By the resolution of $\wedge ^k(\overline {V_n}^{\vee })$ :
  $$ \begin{align*}0\to \mathcal{L}^{k}\to \cdots \to \wedge^{-k-1}(V_{n}^{\vee})\mathcal{L}^{-1}\to \wedge^k(V_{n}^{\vee})\to \wedge^k(\overline{V_{n}}^{\vee})\to 0,\end{align*} $$
  we have
  (3.12) $$ \begin{align} \mathbf{R}q_{n*}(\mathcal{L}_2^k)&=\omega_S^{-k}\otimes \{\cdots \to \mathcal{L}^{-1}\mathbf{L}p_n^*(S^{-k-1}\mathcal{I}_n^{\vee})[1]\to \mathbf{L}p_n^*(S^{-k}\mathcal{I}_n^{\vee}) \} \end{align} $$

By Lemmas 3.8 and 3.9, we have

Corollary 3.10. We have the following formula for $\mathbf {R}(p_n\circ q_n)_*\mathcal {L}_1^{m-1-k}\mathcal {L}_2^k$ :

(3.13) $$ \begin{align} &\mathbf{R}(p_n\circ q_n)_*(\mathcal{L}_1^{m-1-k}\mathcal{L}_2^k)= \\ &\begin{cases} \omega_S^{-k}\otimes \{\wedge^{-m}(\mathcal{I}_n^{\vee})\to \cdots \to \wedge^{-m+k}(\mathcal{I}_n^{\vee})S^{-k}(\mathcal{I}_n^{\vee})\}[m-k] & m\leq k \leq -1 \\ \omega_S^{-k}\otimes \{\wedge^{k}(\mathcal{I}_n)S^{m-k}(\mathcal{I}_n)\to \cdots \to \mathcal{I}_n\otimes S^{m-1}(\mathcal{I}_n)\to S^{m}(\mathcal{I}_n)\}[1-2k] & 1\leq k\leq m.\\ \end{cases} \nonumber\end{align} $$

Remark 3.11. When $m=k<0$ , same as the computation in the Grothendieck group, the complex

$$ \begin{align*}\{\wedge^{-k}(\mathcal{I}_n^{\vee})\to \cdots \to S^{-k}(\mathcal{I}_n^{\vee})\}\end{align*} $$

is quasi-isomorphic to $0$ , and thus, we have

(3.14) $$ \begin{align} \mathbf{R}(p_n\circ q_n)_*\mathcal{L}^{-1}_1\mathcal{L}_2^{k}=0. \end{align} $$

Proof of Theorem 3.2

It follows from Definition 2.4, Lemma 3.8 and Corollary 3.10.

4 The quadrulple/triple moduli spaces and the minimal model program

In this section, we introduce the triple moduli spaces $\mathfrak {Z}_+,\mathfrak {Z}_{-}$ and the quadruple moduli space $\mathfrak {Y}$ which parameterise diagrams:

(4.1)

(4.2)

(4.3)

respectively, of ideal sheaves, where each successive inclusion is colength $1$ and supported at the point indicated on the diagrams. We consider line bundles $\mathcal {L}_{1},\mathcal {L}_{2},\mathcal {L}_{1}',\mathcal {L}_2'$ over triple/quadruple moduli spaces with fibre $\mathcal {I}_{n+1}/\mathcal {I}_{n},\mathcal {I}_n/\mathcal {I}_{n-1},\mathcal {I}_{n+1}/\mathcal {I}_{n}',\mathcal {I}_{n}'/\mathcal {I}_{n-1}$ , respectively.

We consider the Cartesian diagram:

(4.4)

(3.1) is the restriction of (4.4) to the diagonal $\Delta :S^{[n]}\times S\to S^{[n]}\times S^{[n]}\times S\times S$ .

Example 4.1. When $n=1$ , then $\mathcal {I}_{0}=\mathcal {O}_{S}$ , and thus, $\mathfrak {Y}=S^{[1,2]}=Bl_{\Delta _S}(S\times S)$ . The scheme $\mathfrak {Z}_{-}=S\times S$ and $\alpha _{-}$ is the projection morphism of the blow up. The scheme $\mathfrak {Z}_+$ is induced by the Cartesian diagram

(4.5)

and has two irreducible components, such that each one is isomorphic to $Bl_{\Delta }(S\times S)$ .

The main purpose of this section is to compute $\mathbf {R}\alpha _{+*}\mathcal {O}_{\mathfrak {Y}}$ and $\mathbf {R}\alpha _{-*}\mathcal {O}_{\mathfrak {Y}}$ explicitly.

Proposition 4.2. We have the formula

(4.6) $$ \begin{align} \mathbf{R}\alpha_{-*}\mathcal{O}_{\mathfrak{Y}}=\mathcal{O}_{\mathfrak{Z}_-}, \quad \mathbf{R}\alpha_{+*}\mathcal{O}_{\mathfrak{Y}}=\mathcal{O}_{W_0} ,\end{align} $$

where $W_0$ will be defined in Section 4.3.

Proposition 4.2 follows from Proposition 4.7 and Corollary 4.14, which will be proved later in this section. This section would rely on the singularity theory of the minimal model program, which is summarised in Appendix B.

4.1 The geometry of $\mathfrak {Y}$

Theorem 4.3 (Propositions 2.28 and 5.28 of [26]).

The scheme $\mathfrak {Y}$ is smooth of dimension $2n+2$ . The closed embedding:

$$ \begin{align*}\Delta_{\mathfrak{Y}}:S^{[n-1,n,n+1]}\to \mathfrak{Y} \quad (\mathcal{I}_{n-1},\mathcal{I}_n,\mathcal{I}_{n+1},x)\to (\mathcal{I}_{n-1},\mathcal{I}_n,\mathcal{I}_n,\mathcal{I}_{n+1},x,x)\end{align*} $$

is a regular closed subscheme of codimension $1$ . If we abuse the notation to denote $S^{[n-1,n,n+1]}$ by $\Delta _{\mathfrak {Y}}$ , then the morphism of coherent sheaves over $\mathfrak {Y}\times S$ :

$$ \begin{align*}\mathcal{I}_n/\mathcal{I}_{n+1}\to \mathcal{I}_{n-1}/\mathcal{I}_{n+1}\to \mathcal{I}_{n-1}/\mathcal{I}_{n}', \quad \mathcal{I}_n'/\mathcal{I}_{n+1}\to \mathcal{I}_{n-1}/\mathcal{I}_{n+1}\to \mathcal{I}_{n-1}/\mathcal{I}_{n}\end{align*} $$

induce short exact sequences

$$ \begin{align*} 0\to \mathcal{L}_1\to \mathcal{L}_2' \to \mathcal{L}_2'\mathcal{O}_{\Delta_{\mathfrak{Y}}}\to 0 \\ 0\to \mathcal{L}_1'\to \mathcal{L}_2 \to \mathcal{L}_2\mathcal{O}_{\Delta_{\mathfrak{Y}}}\to 0. \end{align*} $$

Moreover, $\mathcal {L}_1\mathcal {L}_2^{\prime -1}=\mathcal {L}_1'\mathcal {L}_2^{-1}=\mathcal {O}(-\Delta _{\mathfrak {Y}})$ . The normal bundle $N_{\mathfrak {Y}/\Delta _{\mathfrak {Y}}}=\mathcal {L}_1^{-1}\mathcal {L}_2$ .

Remark 4.4. From now on, we will abuse the notation to denote $S^{[n-1,n,n+1]}$ by $\Delta _{\mathfrak {Y}}$ .

Lemma 4.5 (Claim 3.8 of [24]).

Let U be the complement of $\Delta _{{\mathfrak {Y}} }$ in ${\mathfrak {Y}}$ . Then $\alpha _+$ and $\alpha _-$ in (4.4) are isomorphisms when restricting to U.

4.2 The geometry of $\mathfrak {Z}_-$

Proposition 4.6. The scheme $\mathfrak {Z}_-$ is an irreducible $2n+2$ dimensional locally complete intersection scheme.

Proof. First we notice that

$$ \begin{align*} \mathfrak{Z}_{-}=S^{[n-1,n]}\times_{S^{[n-1]}}S^{[n-1,n]} \end{align*} $$

has the expected dimension also equal to $2n+2$ . Thus, $\mathfrak {Z}_-$ is a locally complete intersection scheme and, thus, is Cohen-Macaulay.

We recall the Serre’s criterion [34, Tag 033P] for the normality of a variety: to prove that a variety is normal, we only need to show that it satisfies the Serre’s condition S2 (which is always satisfied if the variety is Cohen-Macaulay), and R1 (i.e. the singular locus has at least codimension 1). By Lemma 4.5, $\alpha ^{-1}_-(U)\cong \theta ^{-1}(U)$ is a $2n+2$ -dimensional smooth open subscheme and the complement is the $2n$ -dimensional closed subscheme $S^{[n-1,n]}$ by (4.4). Hence, $\mathfrak {Z}_-$ satisfies R1 and is normal.

Proposition 4.7. Let $K_{\mathfrak {Z}_-}$ and $K_{\mathfrak {Y}}$ be the canonical divisors of $\mathfrak {Z}_-$ and $\mathfrak {Y}$ , respectively. Then

$$ \begin{align*} \alpha_-^*K_{\mathfrak{Z}_-}=K_{\mathfrak{Y}}+\mathcal{O}(\Delta_{\mathfrak{Y}}) \end{align*} $$

and $\mathfrak {Z}_-$ is a canonical singularity. Moreover, we have the formula

$$ \begin{align*} \mathbf{R}\alpha_{-*}(\mathcal{O}_{\mathfrak{Y}})=\mathcal{O}_{\mathfrak{Z}_{-}}. \end{align*} $$

Proof. The complement of $\theta ^{-1}(U)$ in $\mathfrak {Y}$ is $\Delta _{\mathfrak {Y}}=S^{[n-1,n,n+1]}$ , and, thus, there exists $a\in \mathbb {Q}$ , such that

$$ \begin{align*} \alpha_-^*K_{\mathfrak{Z}_-}=K_{\mathfrak{Y}}+a\mathcal{O}(\Delta_{\mathfrak{Y}}). \end{align*} $$

Given two closed points $x,y\in S$ , let $\mathcal {I}_{x}$ and $\mathcal {I}_{y}$ be the ideal sheaf of closed point $x,y$ . We consider

$$ \begin{align*} V_2:=\{(\mathcal{I}_{n-1},x,y)\in S^{[n]}\times S\times S|(\mathcal{I}_{n-1},x)\notin \mathcal{Z}_{n-1} \text{ and } (\mathcal{I}_{n-1},y)\notin \mathcal{Z}_{n-1}\} \end{align*} $$

and regard $V_{2}$ as an open subvariety of $\mathfrak {Z}_{-}$ through the embedding

$$ \begin{align*} (\mathcal{I}_{n-1},x,y)\to (\mathcal{I}_{n-1},\mathcal{I}_{n-1}\cap \mathcal{I}_{x},\mathcal{I}_{n-1}\cap \mathcal{I}_{y}). \end{align*} $$

Let $V_{1}:=\alpha _{-}^{-1}(V_2)$ . We denote $\mathfrak {Y}_{1}$ the quadruple moduli space $\mathfrak {Y}$ when $n=1$ . By Example 4.1, $\mathfrak {Y}_{1}=Bl_{\Delta _{S}}(S\times S)$ and we have the Cartesian diagram:

while the right vertical arrow of the above diagram is the projection morphism of the blow up. Thus, by Lemma B.2, $a=1$ and $\mathfrak {Z}_-$ is a canonical singularity and, hence, a rational singularity by Theorem B.7.

4.3 The geometry of $\mathfrak {Z}_+$

The geometry of $\mathfrak {Z}_{+}$ is more complicated, as it is no longer irreducible. We define $W_0$ as a closed subscheme of $\mathfrak {Z}_{+}$ by

$$ \begin{align*} W_0:=\{(\mathcal{I}_n,\mathcal{I}_{n}',\mathcal{I}_{n+1},x,y)\in \mathfrak{Z}_+|(\mathcal{I}_{n}',y)\in \mathcal{Z}_{n} \text{ and } (\mathcal{I}_{n},x)\in \mathcal{Z}_{n}\}. \end{align*} $$

$W_0$ is the closure of $\beta _+^{-1}(U)$ in $\mathfrak {Z}_{+}$ . We define $W_{1}:=S^{[n,n+1]}$ .

Proposition 4.8. The scheme $\mathfrak {Z}_+$ is a locally complete intersection scheme (and hence, Cohen-Macaulay) of dimension $2n+2$ , with two irreducible components $W_0$ and $W_1$ . $W_0\cap W_1=p_n^{-1}\mathcal {Z}_n$ .

Proof. $W_1$ is $2n+2$ dimensional and the complement of $W_1$ in $\mathfrak {Z}_+$ is $\beta _+^{-1}(U)=\theta ^{-1}(U)$ , which is also $2n+2$ dimensional by Lemma 4.5. Thus, $\mathfrak {Z}_+$ is $2n+2$ dimensional and

$$ \begin{align*}\mathfrak{Z}_{+}=S^{[n,n+1]}\times_{S^{[n+1]}}S^{[n,n+1]}\end{align*} $$

has expected dimension also equal to $2n+2$ . Hence, $\mathfrak {Z}_+$ is a locally complete intersection scheme.

Any closed point of $W_{0}\cap W_1$ corresponds to $(\mathcal {I}_{n},\mathcal {I}_{n+1},x)\in S^{[n,n+1]}$ , such that x has length $\geq 1$ in $\mathcal {O}/\mathcal {I}_n$ and, thus, is in $p_n^{-1}\mathcal {Z}_n$ .

Example 4.9. When $n=2$ , by Example 3.1, $\mathcal {Z}_2=S^{[1,2]}$ . Thus, for any point $(\mathcal {I}_2',\mathcal {I}_2,\mathcal {I}_{3},x,y)\in W_0$ , there exists a unique ideal sheaf $\mathcal {I}_1$ with two short exact sequences

$$ \begin{align*}0\to \mathcal{I}_2'\to \mathcal{I}_1\to k_y\to 0 \quad 0\to \mathcal{I}_2\to \mathcal{I}_1\to k_x\to 0.\end{align*} $$

Thus, $W_0\cong \mathfrak {Y}$ .

$W_0\cap W_1=S^{[1,2,3]}$ by Example 3.1 and is smooth. So $(\mathfrak {Z}_+,0)$ is a semi-snc pair.

Consider the closed subscheme $W_2\subset p_{n}^{-1}\mathcal {Z}_n$

$$ \begin{align*}W_2:=\{(\mathcal{I}_{n},\mathcal{I}_{n+1},x)\in W_1|\text{ the length of } k_x \text{ in } \mathcal{O}/\mathcal{I}_n\geq 3.\}.\end{align*} $$

Lemma 4.10. The schemes $q_n^{-1}(W_2)$ and $W_2$ have dimension less or equal to $2n-1$ .

Proof. The scheme

$$ \begin{align*}S^{[n-2,n-1,n,n+1]}:=S^{[n-2,n-1,n]}\times_{S^{[n-1,n]}}S^{[n-1,n,n+1]}\end{align*} $$

has dimension $2n-1$ by (5.21) of [26]. The image of the projection morphism

$$ \begin{align*}S^{[n-2,n-2,n,n+1]}\to S^{[n,n+1]}\end{align*} $$

is $W_2$ , and the image of the projecton morphism

$$ \begin{align*}S^{[n-2,n-2,n,n+1]}\to S^{[n-1,n,n+1]}\end{align*} $$

is $q_{n}^{-1}(W_2)$ . Hence, $q_n^{-1}(W_2)$ and $W_2$ have dimension less or equal to $2n-1$ .

As the morphism $\alpha _{+}:\mathfrak {Y}\to \mathfrak {Z}_{+}$ factors through $W_0$ , we will abuse the notation to denote the morphism $\alpha _{+}:\mathfrak {Y}\to W_0$ .

Lemma 4.11. The morphism $\alpha _{+}:\mathfrak {Y}\to W_0$ is an isomorphism when restricting to $W_0-W_2$ .

Proof. We denote $\mathfrak {Y}_{2}$ the quadruple moduli space when $n=2$ . Let

$$ \begin{align*}V_3:=\{(\mathcal{I}_{n-2},(\mathcal{I}_1,\mathcal{I}_2,\mathcal{I}_{2}',\mathcal{I}_3,x,y))\in S^{[n-1]}\times \mathfrak{Y}_2|\mathcal{I}_{n-2}+\mathcal{I}_3=\mathcal{O}\}\end{align*} $$

and $V_4:=\alpha _{+}V_3$ . The morphism

$$ \begin{align*}(\mathcal{I}_{n-2},(\mathcal{I}_1,\mathcal{I}_2,\mathcal{I}_{2}',\mathcal{I}_3,x,y))\to (\mathcal{I}_{n-2}\cap\mathcal{I}_1,\mathcal{I}_{n-2}\cap\mathcal{I}_2,\mathcal{I}_{n-2}\cap\mathcal{I}_{2}',\mathcal{I}_{n-2}\cap\mathcal{I}_3,x,y)\end{align*} $$

induces an open emebedding $V_3\subset \mathfrak {Y}$ . By Example 4.9,

$$ \begin{align*}V_3\cong V_4.\end{align*} $$

By Lemma 4.5, $\alpha _{+}$ is also an isomorphism when restricting to $\beta _+^{-1}(U)$ . $W_2$ is the complement of $\beta _+^{-1}(U)\cup V_4$ in $W_0$ .

Let $\overline {W_0}$ be the normalisation of $W_0$ and $\overline {W_0\cap W_1},\overline {W_2}$ the preimage of $W_0\cap W_1$ and $W_2$ in the normalisation. The morphism $\alpha _{+}:\mathfrak {Y}\to W_0$ will factor through $\overline {\alpha _{+}}:\mathfrak {Y}\to \overline {W_0}$ .

Lemma 4.12. The scheme $\overline {W}_0$ is a canonical singularity, and the pair $(\overline {W_0},\overline {W_0\cap W_1})$ is plt.

Proof. The codimension of $\overline {W_2}$ in $\overline {W_0}$ is $3$ , and $\overline {\alpha _{+}}$ is an isomorphism outside of $\overline {W_2}$ . The preimage of $\overline {W_0\cap W_1}$ in $\mathfrak {Y}$ is $\Delta _{\mathfrak {Y}}$ , which is a smooth divisor of $\mathfrak {Y}$ . Hence, $\overline {W}_0$ is a canonical singularity and the pair $(\overline {W_0},\overline {W_0\cap W_1})$ is plt.

Proposition 4.13. The pair $(\mathfrak {Z}_+,0)$ is semi-dlt, and $W_0$ is normal.

Proof. Let

$$ \begin{align*}V_5:=\mathcal{I}_{n-2},(\mathcal{I}_2,\mathcal{I}_{2}',\mathcal{I}_3,x,y))\in S^{[n-1]}\times \mathfrak{Z}_2^{+}|\mathcal{I}_{n-2}+\mathcal{I}_3=\mathcal{O}\},\end{align*} $$

where $\mathfrak {Z}_2^{+}=S^{[2,3]}\times _{S^{[2]}}S^{[2,3]}$ is the triple moduli space when $n=2$ . $V_5$ is an open subscheme of $\mathfrak {Z}_{+}$ . The pair $(V_5,0)$ is a semi-snc by Example 4.9 and so is $(V_5\cup \beta _{+}^{-1}(U),0)$ . $W_2$ is the complement of $V_5\cup \beta _{+}^{-1}(U)$ in $\mathfrak {Z}_{+}$ , and

$$ \begin{align*} codim_{W_2}\mathfrak{Z}_+=3. \end{align*} $$

By Definition B.13, $W_0\cap W_1$ is the conductor subscheme of $\mathfrak {Z}_{+}$ . By Example B.9, $W_0\cap W_1$ is a canonical singularity. By the inversion of adjunction theorem Theorem B.4, the pair $(W_1,W_0\cap W_1)$ is plt and, thus, dlt.

By Lemma 4.12, $(\overline {W_0},\overline {W_0\cap W_1})$ is also a plt pair and, thus, a dlt pair.

By Proposition B.17, $(\mathfrak {Z}_+,0)$ is a semi-dlt pair and $W_0$ is normal.

Corollary 4.14. We have the formula

$$ \begin{align*} \mathbf{R}\alpha_{+*}(\mathcal{O}_{\mathfrak{Y}})=\mathcal{O}_{W_0}. \end{align*} $$

5 The Proof of Theorem 1.1

In this section, we prove Theorem 1.1. Given an integer n, we denote $\iota :S^{[n]}\times S^{[n]}\times S_{1}\times S_{2}\to S^{[n]}\times S^{[n]}\times S_{2}\times S_{1}$ as the morphism which is identity on $S^{[n]}\times S^{[n]}$ but change the order of two copies of S.

Theorem 5.1. Given two integers m and r and let

$$ \begin{align*} e_{m-r}=\mathcal{L}_1^{m-r}\mathcal{O}_{S^{[n,n+1]}}\in D^b(S^{[n]}\times S^{[n+1]}\times S) \\ f_r=\mathcal{L}_1^{k-1}\mathcal{O}_{S^{[n,n+1]}}[1]\in D^b(S^{[n+1]}\times S^{[n]}\times S). \end{align*} $$

Then

1. (1) If $m>0$ , then there are $b_{m,r}^k\in D^b(S^{[n]}\times S^{[n]}\times S\times S)$ for $0\leq k \leq m$ , such that $b_{m,r}^{m}=\iota _{*}e_{m-r}f_r$ and $b_{m,r}^{0}=f_re_{m-r}$ , with exact triangles

   $$ \begin{align*} \mathfrak{B}_{m,r}^{k}:\mathbf{R}\Delta_*(h_{m,k}^{+})[-1]\to b_{m,r}^{k}\to b_{m,r}^{k+1}\to \mathbf{R}\Delta_*(h_{m,k}^{+}) \end{align*} $$

2. (2) If $m<0$ , then there are $c_{m,r}^r\in D^{b}(S^{[n]}\times S^{[n]}\times S \times S)$ for $m\leq k\leq 0$ , such that $c_{m,r}^{m}=\iota _{*}e_{m-r}f_r$ and $c_{m,r}^{0}=f_re_{m-r}$ , and exact triangles

   $$ \begin{align*} \mathfrak{C}_{m,r}^k:\mathbf{R}\Delta_*(h_{m,k}^-)[-1]\to c_{m,r}^{k-1}\to c_{m,r}^{k}\to \mathbf{R}\Delta_*(h_{m,k}^-) \end{align*} $$

3. (3) If $m=0$ , then there is an explicit isomorphism $f_re_{-r}=e_rf_{-r}\oplus \mathcal {O}_{\Delta }[1]$ .

Theorem 1.1 follows from Theorems 3.2 and 5.1. We will prove Theorem 5.1 later in this section.

5.1 $\iota _{*}e_{m-r}f_r$ and $f_re_{m-r}$ revisited

As $\mathfrak {Z}_{-}$ and $\mathfrak {Z}_{+}$ are both Cohen-Macaulay of expected dimension, we have the following formula:

$$ \begin{align*} \iota_{*}e_{m-r}f_{r}=\mathbf{R}\beta_{-*}(\mathcal{L}_2^{\prime m-r}\mathcal{L}_2^{r-1}\mathcal{O}_{\mathfrak{Z}_{-}})[1]\in D^b(S^{[n]}\times S^{[n]}\times S \times S) \\ f_re_{m-r}=\mathbf{R}\beta_{+*}(\mathcal{L}_1^{m-r}\mathcal{L}_1^{\prime r-1}\mathcal{O}_{\mathfrak{Z}_{+}})[1]\in D^b(S^{[n]}\times S^{[n]}\times S \times S). \end{align*} $$

By Proposition 4.6, $\mathbf {R}\alpha _{-*}\mathcal {O}_{\mathfrak {Y}}=\mathcal {O}_{\mathfrak {Z}_-}$ , and by Theorem 4.3

(5.1) $$ \begin{align} \iota_{*}e_{m-r}f_r&=\mathbf{R}\theta_*(\mathcal{L}_2^{\prime m-r}\mathcal{L}_2^{r-1}\mathcal{O}_{\mathfrak{Y}})[1] \nonumber \\ &=\mathbf{R}\theta_*(\mathcal{L}_1^{\prime m-r}\mathcal{L}_1^{r-1}\mathcal{O}((m-1)\Delta_{\mathfrak{Y}}))[1]. \end{align} $$

5.2 $a_{m,r}^{k}$ and $\mathfrak {A}_{m,r}^k$

We recall the short exact sequence

(5.2) $$ \begin{align} 0\to \mathcal{O}_{\mathfrak{Y}}(-\Delta_{\mathfrak{Y}})\to \mathcal{O}_{\mathfrak{Y}} \to \mathcal{O}_{\Delta_{\mathfrak{Y}}}\to 0. \end{align} $$

Definition 5.2. For any integer r, we define

$$ \begin{align*}a_{m,r}^{k}=\mathbf{R}\theta_{*}(\mathcal{L}_{1}^{\prime m-r}\mathcal{L}_{1}^{r-1}\mathcal{O}(k\Delta_{\mathfrak{Y}}))[1].\end{align*} $$

We define the exact triangles

(5.3) $$ \begin{align} \mathfrak{A}_{m,r}^k :a_{m,r}^{k-1}\to a_{m,r}^{k} \to \mathbf{R}\Delta_*(\mathbf{R}(p_n\circ q_n)_*\mathcal{L}_{1}^{m-1-k}\mathcal{L}_{2}^k)[1] \to a_{m,r}^{k-1}[1] \end{align} $$

by applying the functor $\mathbf {R}\theta _{*}(-\otimes \mathcal {L}_{1}^{\prime m-r}\mathcal {L}_{1}^{r-1}\mathcal {O}(k\Delta _{\mathfrak {Y}}))[1]$ to (5.2).

By (5.1), $\iota _{*}e_{m-r}f_r\cong a_{m,r}^{m-1}$ . Moreover, when $m<0$ , by (3.14), $\mathbf {R}(p_n\circ q_n)_*\mathcal {L}_{1}^{-1}\mathcal {L}_{2}^m=0$ and $\iota _{*}e_{m-r}f_r\cong a_{m,r}^{m-1}\cong a_{m,r}^m$ .

5.3 $\mathfrak {B}_{m,r}^{\prime 0}$ and $\mathfrak {C}_{m,r}^{\prime 0}$

By Example B.9, $\mathbf {R}\alpha _{+*}\mathcal {O}_{\Delta _{\mathfrak {Y}}}=\mathcal {O}_{W_0\cap W_1}$ . Taking the $\mathbf {R}\alpha _{+*}$ functor to the short exact sequence (5.2), we have the short exact sequence

$$ \begin{align*} 0\to \mathbf{R}\alpha_{+*}\mathcal{O}(-\Delta_{\mathfrak{Y}})\to \mathcal{O}_{W_0}\to \mathcal{O}_{W_0\cap W_1}\to 0. \end{align*} $$

Recalling the short exact sequence in Corollary 3.6:

$$ \begin{align*}0\to \mathcal{L}\otimes \mathcal{O}_{W_1}\to \mathcal{O}_{W_1}\to \mathcal{O}_{W_0\cap W_1}\to 0.\end{align*} $$

We have the following commutative diagram:

where all rows and columns are short exact sequences. Thus, we have short exact sequences:

(5.4) $$ \begin{align} 0\to \mathbf{R}\alpha_{+*}\mathcal{O}(-\Delta_{\mathfrak{Y}})\to \mathcal{O}_{\mathfrak{Z}_+} \to \mathcal{O}_{W_1}\to 0 \end{align} $$

(5.5) $$ \begin{align} 0\to \mathcal{L}_1\mathcal{O}_{S^{[n,n+1]}}\to \mathcal{O}_{\mathfrak{Z}_{+}} \to \mathcal{O}_{W_0}\to 0. \end{align} $$

Taking the functor $\mathbf {R}\beta _{+*}(-\otimes \mathcal {L}_{1}^{\prime m-r}\mathcal {L}_{1}^{r-1})[1]$ to (5.4) and (5.5), respectively, we get the exact triangles

(5.6) $$ \begin{align} & \mathfrak{C}_{m,r}^{\prime 0}:a_{m,r}^{-1}\to f_{r}e_{m-r} \to \mathbf{R}\Delta_*(\mathbf{R}p_{n*} \mathcal{L}^{m-1})[1] \to a_{m,r}^{-1}[1] \end{align} $$

(5.7) $$ \begin{align} & \mathfrak{B}_{m,r}^{\prime 0}: \mathbf{R}\Delta_*(\mathbf{R}p_{n*}\mathcal{L}^{m})[1] \to f_{r}e_{m-r} \to a_{m,r}^0 \to \mathbf{R}\Delta_*(\mathbf{R}p_{n*}\mathcal{L}^{m})[2]. \end{align} $$

5.4 $\mathfrak {B}_{m,r}^k$ and $\mathfrak {C}_{m,r}^k$

Definition 5.3. When $m>0$ , we define $b_{m.r}^k\in D^{b}(S^{[n]}\times S^{[n]}\times S\times S)$ and natural transforms $\mathfrak {B}_{m,r}^k:b_{m,r}^{k}\to b_{m,r}^{k+1}$ by

(5.8) $$ \begin{align} b_{m,r}^k:= \begin{cases} f_{r}e_{m-r} & k=0\\ a_{m,r}^{k-1} & 1\leq k\leq m \end{cases} \end{align} $$

(5.9) $$ \begin{align} \mathfrak{B}_{m,r}^k:= \begin{cases} \mathfrak{B}_{m,r}^{\prime 0} & k=0\\ \mathfrak{A}_{m,r}^{k}& 1\leq k\leq m-1. \end{cases} \end{align} $$

We have $b_{m,r}^{0}=f_re_{m-r}$ and $b_{m,r}^{m}=\iota _{*}e_{m-r}f_r$ .

When $m<0$ , we define $c_{m,r}^k\in D^{b}(S^{[n]}\times S^{[n]}\times S\times S)$ and $\mathfrak {C}_{m,r}^k:c_{m,r}^{k-1}\to c_{m,r}^{k}$ by

(5.10) $$ \begin{align} c_{m,r}^k= \begin{cases} f_{r}e_{m-r} & k=0\\ a_{m,r}^k & m\leq k\leq -1 \end{cases} \end{align} $$

(5.11) $$ \begin{align} \mathfrak{C}_{m,r}^k= \begin{cases} \mathfrak{C}_{m,r}^{\prime 0} & k=0 \\ \mathfrak{A}_{m,r}^k & m+1 \leq k \leq -1. \end{cases} \end{align} $$

We have $f_re_{m-r}=c_{m,r}^{0}$ and $\iota _{*}e_{m-r}f_r=c_{m,r}^{m}$ .

Proof of Theorem 5.1

When $m>0$ , we only need to prove that the cone of $\mathfrak {B}_{m,r}^k$ is $\mathbf {R}\Delta _*(h_{m,k}^+)$ and the cone of $\mathfrak {C}_{m,r}^k$ is $\mathbf {R}\Delta _*(h_{m,k}^-)$ . It follows from (5.3), (5.6) and (5.7).

When $m=0$ ,

$$ \begin{align*} e_{-r}f_r=\mathbf{R}\beta_{+*}(\mathbf{R}\alpha_{+*}(\mathcal{L}_1^{\prime -r}\mathcal{L}_1^{r-1}\mathcal{O}(-\Delta_{\mathfrak{Y}}))). \end{align*} $$

We have the short exact sequence:

(5.12) $$ \begin{align} 0\to \mathcal{L}_1^{\prime -r}\mathcal{L}_1^{r-1}\mathcal{O}_{\mathfrak{Z}_+}\to \mathcal{L}_1^{\prime -r}\mathcal{L}_1^{r-1}(\mathcal{O}_{W_0}\oplus\mathcal{O}_{W_1})\to \mathcal{L}^{-1}\mathcal{O}_{W_0\cap W_1}\to 0, \end{align} $$

and

$$ \begin{align*} \mathbf{R}\beta_{+*}(\mathcal{L}^{-1}\mathcal{O}_{W_0\cap W_1})&=\{\mathbf{R}\beta_{+*}(\mathcal{O}_{W_1}\to \mathcal{L}^{-1}\mathcal{O}_{W_1})\} \\ &=\mathbf{R}\Delta_{*}\{\mathcal{O}_{S^{[n]}\times S}\to \mathcal{O}_{S^{[n]}\times S}\} \text{ by Lemma~3.8 }\\ &=0. \end{align*} $$

By (5.4) and (5.12), we have isomorphisms

$$ \begin{align*} & f_{r}e_{-r}\cong \mathbf{R}\beta_{+*}\mathcal{L}_1^{\prime -r}\mathcal{L}_1^{r-1}(\mathcal{O}_{W_0})[1]\oplus\mathcal{O}_{\Delta}[1] \\ & e_{-r}f_{r}\cong \mathbf{R}\beta_{+*}\mathcal{L}_1^{\prime -r}\mathcal{L}_1^{r-1}(\mathcal{O}_{W_0})[1].\\[-40pt] \end{align*} $$

6 Extension classes between $h_{m,k}^{\pm }$

When $m>0$ , the composition of $\mathfrak {B}_{m,r}^k$ and $\mathfrak {B}_{m,r}^{k+1}$ in Theorem 5.1 induces an extension of $\mathbf {R}\Delta _*h_{m,k-1}^{+}$ and $\mathbf {R}\Delta _*h_{m,k}^{+}$ , that is an extension class of

$$ \begin{align*}Hom(\mathbf{R}\Delta_*h_{m,k}^{+},\mathbf{R}\Delta_*h_{m,k-1}^{+}[1]).\end{align*} $$

Similarly, when $m<0$ , there is also an extension class of

$$ \begin{align*}Hom(\mathbf{R}\Delta_*h_{m,k}^{-},\mathbf{R}\Delta_*h_{m,k-1}^{-}[1])\end{align*} $$

which relies on $m,k,r$ . The purpose of this section is to study the above extension classes. First we recall the Hochschild-Kostant-Rosenberg theorem (short for HKR theorem). Let $f:X\to Y$ be a regular embedding of smooth varieties. On X, we have the following short exact sequence

$$ \begin{align*}0\to N_{Y/X}^{{\vee}}\xrightarrow{\iota_{Y/X}}T^{*}Y|_{X}\to T^{*}X\to 0.\end{align*} $$

Let $g:T^{*}Y\to N_{Y/X}^{{\vee }}$ be a splitting of the above short exact sequence, that is $g{\circ }\iota _{Y/X}=id$ .

Theorem 6.1 (HKR isomorphism, Theorem 1.4 and Section 1.11 of [1], see Theorem C.10 for the precise formulation).

The splitting g induces a canonical isomorphism

$$ \begin{align*}\tau_{N_{Y/X}^{{\vee}}}^{F,G}:\mathbf{R}Hom_{X}(F{\otimes}^{\mathbf{L}}(\bigoplus_{j=0}^{\infty}{\wedge}^{j}N_{Y/X}^{{\vee}}),G){\cong}\mathbf{R}Hom_{Y}(\mathbf{R}f_{*}F,\mathbf{R}f_{*}G)\end{align*} $$

which is functorial respect to $F,G\in D(X)$ .

Remark 6.2. The isomorphism $\tau _{N_{X/Y}^{{\vee }}}^{F,G}$ depends on the choice of the splitting g. For a different choice of g, the difference of the isomorphism $\tau _{N_{X/Y}^{{\vee }}}^{F,G}$ is represented by a transition matrix in Proposition 3.7 of [13].

Now we consider the diagonal embedding $\Delta :S^{[n]}{\times }S\to S^{[n]}{\times }S^{[n]}{\times }S{\times }S$ . Then $T^{*}(S^{[n]}{\times }S^{[n]}{\times }S{\times }S)|_{\Delta }=T^{*}(S^{[n]}{\times }S){\oplus }T^{*}(S^{[n]}{\times }S)$ , where the conormal bundle is

$$ \begin{align*}T^{*}(S^{[n]}{\times}S){\cong}\{(x,-x)\in T^{*}(S^{[n]}{\times}S){\oplus}T^{*}(S^{[n]}{\times}S)\}.\end{align*} $$

We take the split

(6.1) $$ \begin{align} (T^{*}\Delta)^{-1}:T^{*}(S^{[n]}{\times}S){\oplus}T^{*}(S^{[n]}{\times}S)\to T^{*}(S^{[n]}{\times}S) \end{align} $$

which maps $(a,b)$ to $(\frac {1}{2}(a-b),\frac {1}{2}(b-a))$ and, hence, induce isomorphisms

$$ \begin{align*} Hom(\mathbf{R}\Delta_*h_{m,k}^{+},\mathbf{R}\Delta_*h_{m,k-1}^{+}[1])=\bigoplus_{j=0}^{\infty} Hom_{S^{[n]}{\times}S}(h_{m,k}^+\otimes \wedge^jT^*(S^{[n]}\times S),h_{m,k-1}^+[1-j]),\\ Hom(\mathbf{R}\Delta_*h_{m,k}^{-},\mathbf{R}\Delta_*h_{m,k-1}^{-}[1])=\bigoplus_{j=0}^{\infty} Hom_{S^{[n]}{\times}S}(h_{m,k}^{-}\otimes \wedge^jT^*(S^{[n]}\times S),h_{m,k-1}^{-}[1-j]). \end{align*} $$

In this section, we will be explicitly computing the above extension classes and prove that

Proposition 6.3. The extension class in $Hom(\mathbf {R}\Delta _*h_{m,k}^{+},\mathbf {R}\Delta _*h_{m,k-1}^{+}[1])$ is

$$ \begin{align*}(\mathfrak{F}_{m,k,r}^{+},\mathfrak{H}_{m,k}^{+},0,0,\cdots)\end{align*} $$

and the extension class in $Hom(\mathbf {R}\Delta _*h_{m,k}^{-},\mathbf {R}\Delta _*h_{m,k-1}^{-}[1])$ is

$$ \begin{align*}(\mathfrak{F}_{m,k,r}^{-},\mathfrak{H}_{m,k}^-,0,0,\cdots),\end{align*} $$

where $\mathfrak {F}_{m,k,r}^{+}$ , $\mathfrak {F}_{m,k,r}^{-}$ , $\mathfrak {H}_{m,k}^{+}$ and $\mathfrak {H}_{m,k}^{-}$ will be defined in Definition 6.7. The classes $\mathfrak {H}_{m,k}^{+}$ and $\mathfrak {H}_{m,k}^{-}$ only depend on $m,k$ but not r.

Proposition 6.3 will be proved later in this section.

6.1 A morphism $s: \mathcal {L}_{1}^{-1} \mathcal {L}_{2} \to (p_n\circ q_n)^{*}T(S^{[n]}\times S)$

By restricting $\theta :\mathfrak {Y}\to S^{[n]}\times S^{[n]}\times S\times S$ to the diagonal of $S^{[n]}\times S^{[n]}\times S\times S$ , we get the Cartesian diagram:

(6.2)

and the normal bundle of $S^{[n]}\times S$ in $S^{[n]}\times S^{[n]}\times S\times S$ is $T(S^{[n]}\times S)$ . Thus, the diagram (6.2) induces a morphism of short exact sequences:

(6.3)

where $N_{\mathfrak {Y}/\Delta _{\mathfrak {Y}}}{\cong }\mathcal {L}_1^{-1}\mathcal {L}_2$ . The purpose of this subsection is to construct a split $\lambda :{\mathcal {L}}_{1}^{-1}{\mathcal {L}}_{2}\to T{\mathfrak {Y}}|_{S^{n-1,n,n+1}}$ which is compatible with the splitting of tangent bundle (or cotangent bundle) in (6.1) and give an explicit formulation of $s'$ in Proposition 6.6. First we recall the description of the tangent space of $S^{[n-1,n,n+1]}$ and $\mathfrak {Y}$ in Proposition 5.28 of [26]: we consider a closed point $t\in S^{[n-1,n,n+1]}$ corresponds to two short exact sequences of coherent sheaves on S:

$$ \begin{align*} 0\to \mathcal{I}_{n}\xrightarrow{i_n} \mathcal{I}_{n-1}\xrightarrow{j_n} k_x \to 0 & \quad 0\to \mathcal{I}_{n+1}\xrightarrow{i_{n+1}} \mathcal{I}_{n}\xrightarrow{j_{n+1}} k_x \to 0. \end{align*} $$

Let V be the vector space of pairs $\{(w_0,w_1)\in Ext^1(\mathcal {I}_{n-1},\mathcal {I}_{n-1})\oplus Ext^1(\mathcal {I}_{n},\mathcal {I}_n)\}$ , such that $w_0,w_1$ map to the same element of $Ext^1(\mathcal {I}_n,\mathcal {I}_{n-1})$ . By the proof of Proposition 5.28 of [26], elements in V are in 1-1 correspondence with the commutative diagrams of short exact sequences:

and, thus, induce a morphism $dp_S^1:V\to Ext^1(k_x,k_x)$ . Let $V'$ be the vector space of pairs

$$ \begin{align*} \{(w_1',w_2)\in Ext^1(\mathcal{I}_{n},\mathcal{I}_n)\oplus Ext^1(\mathcal{I}_{n+1},\mathcal{I}_{n+1})\}, \end{align*} $$

such that $w_1',w_2$ map to the same element of $Ext^1(\mathcal {I}_{n},\mathcal {I}_{n+1})$ . Then elements in $V'$ are in 1-1 correspondence with the commutative diagrams of short exact sequences:

and, thus, also induces a morphism $dp_S^2:V'\to Ext^1(k_x,k_x)$ .

Consider the natural morphism:

$$ \begin{align*} \hat{s}_1: Hom(\mathcal{I}_n,k_x)\otimes Ext^1(k_x,\mathcal{I}_n)\to Ext^1(\mathcal{I}_n,\mathcal{I}_n) \\ \hat{s}_2: Hom(\mathcal{I}_n,k_x)\otimes Ext^1(k_x,\mathcal{I}_n)\to Ext^1(k_x,k_x), \end{align*} $$

and the short exact sequences

$$ \begin{align*} Hom(\mathcal{I}_n,k_x)\xrightarrow{\hat{s}_1(-\otimes r_2)} Ext^1(\mathcal{I}_n,\mathcal{I}_n)\to Ext^1(\mathcal{I}_n,\mathcal{I}_{n-1}) \\ Ext^{1}(k_x,\mathcal{I}_n)\xrightarrow{\hat{s}_1(r_1\otimes -)}Ext^1(\mathcal{I}_n,\mathcal{I}_n)\to Ext^1(\mathcal{I}_{n+1},\mathcal{I}_n). \end{align*} $$

For any element $u\in Hom(\mathcal {I}_n,k_x)$ , $(\hat {s}_1(u\otimes r_2),0)\in V$ . Moreover, by diagram chasing (we left it to interested readers)

(6.4) $$ \begin{align} dp_S^1(\hat{s}_1(u\otimes r_2),0)=\hat{s}_2(u\otimes r_2). \end{align} $$

Similarly, for any element $v\in Ext^1(k_x,\mathcal {I}_n)$ , $(0,\hat {s}_1(r_1\otimes v))\in V'$ and

(6.5) $$ \begin{align} dp_S^2(0,\hat{s}_1(r_1\otimes v))=\hat{s}_2(r_1\otimes v). \end{align} $$

Lemma 6.4 (Proposition 5.28 of [26]).

The tangent space $T_{t}\mathfrak {Y}$ is the space of $(w_0,w_1,w_1',w_2,u,v)$ in

$$ \begin{align*}&Ext^1(\mathcal{I}_{n-1},\mathcal{I}_{n-1})\oplus Ext^{1}(\mathcal{I}_n,\mathcal{I}_n)\oplus Ext^{1}(\mathcal{I}_n,\mathcal{I}_n)\oplus Ext^{1}(\mathcal{I}_{n+1},\mathcal{I}_{n+1})\oplus Ext^1(k_x,k_x)\\&\quad\oplus Ext^1(k_x,k_x),\end{align*} $$

such that

1. (1) $(w_0,w_1)$ and $(w_0,w_1')$ are in V.

2. (2) $(w_1,w_2)$ and $(w_1',w_2)$ are in $V'$ .

3. (3) $dp_S^1(w_0,w_1)=dp_S^2(w_1',w_2)=u$ ,

4. (4) $dp_S^1(w_0,w_1')=dp_S^2(w_1,w_2)=v$ .

The tangent space $T_{t}S^{[n-1,n,n+1]}$ consists of subspace of $T_{t}\mathfrak {Y}$ , such that $w_{1}=w_{1}'$ and $u=v$ .

Corollary 6.5. The morphism $\lambda _{t}:T_{t}{\mathfrak {Y}}\to T_{t}{\mathfrak {Y}}:(w_0,w_1,w_1',w_2,u,v)\to (0,w_1'-w_1,w_1'-w_1,0,u-v,v-u)$ induces a splitting $\lambda :{\mathcal {L}}_{1}^{-1}{\mathcal {L}}_{2}\to T{\mathfrak {Y}}|_{S^{[n-1,n,n+1]}}$ which is compatible with (6.1). Moreover, under the splitting $\lambda $ , ${\mathcal {L}}_{1}{\mathcal {L}}_{2}^{-1}|_{t}$ is

$$ \begin{align*}\{(w,v)\in Ext^{1}({\mathcal{I}}_{n},{\mathcal{I}}_{n}){\oplus}Ext^{1}(k_{x},k_{x})|(0,w)\in V,(w,0)\in V',dp_{S}^{1}(0,w)=dp_{S}^{2}(w,0)=u\}.\end{align*} $$

Proof. If $(w_0,w_1,w_1',w_2,u,v)$ is in the tangent space, then so is

$$ \begin{align*}\frac{(2w_0,w_1+w_1',w_1+w_1',2w_2,u+v,u+v)}{2} \text{ and } (0,w_1'-w_1,w_1-w_1',0,u-v,v-u).\end{align*} $$

Hence, the tangent space $T_t\mathfrak {Y}$ decomposes into a direct sum of two subspaces: the subspace $w_1=w_1'$ , which is $T_tS^{[n-1,n,n+1]}$ , and the subspace N, which consists of elements $(w_1,u)\in Ext^{1}(\mathcal {I}_n,\mathcal {I}_n)\oplus Ext^{1}(k_x,k_x)$ , such that

1. (1) $w_1$ maps to $0$ in $Ext^{1}(\mathcal {I}_{n}, \mathcal {I}_{n-1})$ and $Ext^{1}(\mathcal {I}_{n+1}, \mathcal {I}_{n}),$

2. (2) $dp_S^1(w_1,0)=dp_S^2(0,w_1)=u.$

We define the morphism $\lambda _{t}$ just to be the projection to N, and it induces a splitting $\lambda :{\mathcal {L}}_{1}^{-1}{\mathcal {L}}_{2}\to T{\mathfrak {Y}}|_{S^{[n-1,n,n+1]}}$ .

Now we give an explicit description of $s'$ . We first construct two canonical morphisms:

(6.6) $$ \begin{align} s_1:\mathcal{L}_1^{-1}\mathcal{L}_2\to (p_n\circ q_n)^*TS^{[n]}\quad s_2:\mathcal{L}_1^{-1}\mathcal{L}_2\to (p_n\circ q_n)^*(TS). \end{align} $$

Let $\pi :S^{[n-1,n,n+1]}\times S\to S^{[n-1,n,n+1]}$ be the projection morphism and $\Gamma $ be the graph of the projection map to S. Then there exists short exact sequences:

$$ \begin{align*}0\to \mathcal{I}_{n+1}\to \mathcal{I}_n\to \pi^*\mathcal{L}_{1}\otimes \mathcal{O}_{\Gamma}\to 0\quad 0\to \mathcal{I}_{n}\to \mathcal{I}_{n-1}\to \pi^*\mathcal{L}_{2}\otimes \mathcal{O}_{\Gamma}\to 0\end{align*} $$

which induce two global sections:

$$ \begin{align*} r_1:\mathcal{L}_1^{-1}\to \mathcal{H}om_{\pi}(\mathcal{I}_n,\mathcal{O}_{\Gamma}), \quad r_2:\mathcal{L}_2\to \mathcal{E}xt^1_{\pi}(\mathcal{O}_{\Gamma},\mathcal{I}_{n}). \end{align*} $$

Let $\text {proj}_n:S^{[n]}\times S\to S^{[n]}$ be the projection map and $\Gamma _n$ be the graph of the projection map from $S^{[n]}\times S\to S$ . Then

$$ \begin{align*} (p_n\circ q_n)^*TS^{[n]}&=(p_n\circ q_n)^*\mathcal{E}xt_{\text{proj}_n}^1(\mathcal{I}_n,\mathcal{I}_n)\\ &=\mathcal{E}xt^1_{\pi}(\mathcal{I}_n,\mathcal{I}_n)\\ (p_n\circ q_n)^*TS&=\mathcal{E}xt^1_{\pi}(\mathcal{O}_{\Gamma},\mathcal{O}_{\Gamma}) \end{align*} $$

by the flat base change theorem. With the composition of the following two natural homomorphisms:

$$ \begin{align*} \mathcal{E}xt^1_{\pi}(\mathcal{O}_{\Gamma},\mathcal{I}_n)\otimes \mathcal{H}om_{\pi}(\mathcal{I}_n, \mathcal{O}_{\Gamma}) \to \mathcal{E}xt^1_{\pi}(\mathcal{I}_n,\mathcal{I}_n)=(p_n\circ q_n)^*TS^{[n]} \\ \mathcal{H}om_{\pi}(\mathcal{I}_n,\mathcal{O}_{\Gamma})\otimes \mathcal{E}xt^1_{\pi}(\mathcal{O}_{\Gamma},\mathcal{I}_n)\to \mathcal{E}xt^1_{\pi}(\mathcal{O}_{\Gamma},\mathcal{O}_{\Gamma})=(p_n\circ q_n)^*TS, \end{align*} $$

we get the two canonical morphisms in (6.6)

$$ \begin{align*} s_1:\mathcal{L}_1^{-1}\mathcal{L}_2\to (p_n\circ q_n)^*TS^{[n]}\quad s_2:\mathcal{L}_1^{-1}\mathcal{L}_2\to (p_n\circ q_n)^*(TS). \end{align*} $$

Let

$$ \begin{align*} s=(s_1,s_2):\mathcal{L}_1^{-1}\mathcal{L}_2\to (p_n\circ q_n)^*T(S^{[n]}\times S). \end{align*} $$

Proposition 6.6. The morphism $s'$ in (6.3) coincides with s in (6.6).

Proof. A closed point t on $S^{[n-1,n,n+1]}$ corresponds to two short exact sequences of coherent sheaves on S:

$$ \begin{align*} 0\to \mathcal{I}_{n}\xrightarrow{i_n} \mathcal{I}_{n-1}\xrightarrow{j_n} k_x \to 0 & \quad 0\to \mathcal{I}_{n+1}\xrightarrow{i_{n+1}} \mathcal{I}_{n}\xrightarrow{j_{n+1}} k_x \to 0, \end{align*} $$

which induces $r_1\in Hom(\mathcal {I}_n,k_x)$ and $r_2\in Ext^1(k_x,\mathcal {I}_n)$ . The image of s is $(\hat {s}_1(r_1\otimes r_2),\hat {s}_2(r_1\otimes r_2))$ , and we only need to prove that it is also the image of N in the proof of Corollary 6.5. It follows from the fact that

$$ \begin{align*}dp_S^1(\hat{s}_1(r_1\otimes r_2),0)=dp_S^2(0,\hat{s}_1(r_1\otimes r_2))=\hat{s}_2(r_1\otimes r_2)\end{align*} $$

by (6.4), (6.5).

6.2 The definition of $\mathfrak {F}_{m,k,r}^{+},\mathfrak {H}_{m,k}^{\pm }$

We recall the Cartesian diagram (6.2)

Consider the dual of s

$$ \begin{align*}s^{\vee}:(p_n\circ q_n)^*T^*(S^{[n]}\times S)\to \mathcal{L}_1\mathcal{L}_2^{-1}\end{align*} $$

and the splitting $\lambda ^{{\vee }}$ in Corollary 6.5. We define

(6.7) $$ \begin{align} \tau_{m}^k:\mathbf{R}(p_n\circ q_n)_*\mathcal{L}_1^{m-1-k}\mathcal{L}_2^k\otimes^{L} T^*(S^{[n]}\times S)\to \mathbf{R}(p_n\circ q_n)_*\mathcal{L}_1^{m-k}\mathcal{L}_2^{k-1} \end{align} $$

as $\mathbf {R}(p_{n}{\circ }q_{n})_{*}(s^{{\vee }}{\otimes }{\mathcal {L}}_{1}^{m-k-1}{\mathcal {L}}_{2}^{k})$ .

On the other hand, we recall the definition of Definition 5.2

$$ \begin{align*}a_{m,r}^{k}=\mathbf{R}\theta_{*}(\mathcal{L}_{1}^{\prime m-r}\mathcal{L}_{1}^{r-1}\mathcal{O}(k\Delta_{\mathfrak{Y}}))[1].\end{align*} $$

By Lemma C.8, the splitting $\lambda $ identify an isomorphism between the first order infinitesimal neighborhood of $\Delta _{{\mathfrak {Y}}}$ in ${\mathfrak {Y}}$ with $Spec_{S^{[n-1,n,n+1]}}({\mathcal {O}}_{S^{[n-1,n,n+1]}}{\oplus }{\mathcal {L}}_{1}{\mathcal {L}}_{2}^{-1})$ . We denote $\alpha _{m,r}^{k}$ as the Bass-Quillen class (see Appendix C.3 for the definition) of $a_{m,r}^{k}$ when restricting to $Spec_{S^{[n-1,n,n+1]}}({\mathcal {O}}_{S^{[n-1,n,n+1]}}{\oplus }{\mathcal {L}}_{1}{\mathcal {L}}_{2}^{-1})$ .

By Lemma 3.9,

$$ \begin{align*}Rq_{n*}(\mathcal{L}_1^{m-1})=\{\mathcal{L}^m\to \mathcal{L}^{m-1}\}\end{align*} $$

is the cone of $\mathcal {L}^m$ to $\mathcal {L}^{m-1}$ . Hence, we have exact triangles

(6.8) $$ \begin{align} Rp_{n*}\mathcal{L}^m\to Rp_{n*}\mathcal{L}^{m-1}\xrightarrow{\mu_m} R(p_n\circ q_n)_*(\mathcal{L}_1^{m-1}) \xrightarrow{\nu_m}Rp_{n*}\mathcal{L}^m[1]. \end{align} $$

Definition 6.7. When $m\geq k> 0$ , we define

$$ \begin{align*}\mathfrak{H}_{m,k}^+:h_{m,k}^+\otimes T^*(S^{[n]}\times S)\to h_{m,k-1}^+\end{align*} $$

by

(6.9) $$ \begin{align} \mathfrak{H}_{m,k}^+= \begin{cases} \tau_{m,k} & k>1 \\ \nu_m\circ \tau_{m,0}& k=1, \end{cases} \end{align} $$

and when $m<k\leq 0$ , we define

$$ \begin{align*}\mathfrak{H}_{m,k}^-:h_{m,k}^-\otimes T^*(S^{[n]}\times S)\to h_{m,k-1}^-\end{align*} $$

by

(6.10) $$ \begin{align} \mathfrak{H}_{m,k}^-= \begin{cases} \tau_{m,k} & k<0 \\ \tau_{m,0}\circ \mu_m & k=0. \end{cases} \end{align} $$

Definition 6.8. When $m\geq k> 0$ and r is an integer, we define

$$ \begin{align*}\mathfrak{F}_{m,k}^+:h_{m,k}^+\otimes T^*(S^{[n]}\times S)\to h_{m,k-1}^+\end{align*} $$

by

(6.11) $$ \begin{align} \mathfrak{F}_{m,k}^+= \begin{cases} \alpha_{m,k} & k>1 \\ \nu_m\circ \alpha_{m,0}& k=1, \end{cases} \end{align} $$

and when $m<k\leq 0$ , we define

$$ \begin{align*}\mathfrak{F}_{m,k}^-:h_{m,k}^-\otimes T^*(S^{[n]}\times S)\to h_{m,k-1}^-\end{align*} $$

by

(6.12) $$ \begin{align} \mathfrak{F}_{m,k}^-= \begin{cases} \alpha_{m,k} & k<0 \\ \alpha_{m,0}\circ \mu_m & k=0. \end{cases} \end{align} $$

Proof of Proposition 6.3

We only compute the postive part of the extension and only consider the $k>1$ and the case $k=1$ follows from the (5.5). The extension class is induced from the short exact sequence on $\mathfrak {Y}$ :

$$ \begin{align*}0\to {\mathcal{L}}_{1}{\mathcal{L}}_{2}^{-1}a_{m,k}^{r}\mathcal{O}_{\Delta_{\mathfrak{Y}}}\to a_{m,k}^{r}\mathcal{O}_{2\Delta_{\mathfrak{Y}}}\to a_{m,k}^{r}\mathcal{O}_{\Delta_{\mathfrak{Y}}}\to 0,\end{align*} $$

where $\mathcal {O}_{2\Delta _{\mathfrak {Y}}}$ is the cokernel of $\mathcal {O}(-2\Delta _{\mathfrak {Y}})\to \mathcal {O}_{\mathfrak {Y}}$ . Hence, we obtain the extension formula through Theorem C.11.