ODD RANK VECTOR BUNDLES IN ETA-PERIODIC MOTIVIC HOMOTOPY THEORY

Abstract

We observe that, in the eta-periodic motivic stable homotopy category, odd rank vector bundles behave to some extent as if they had a nowhere vanishing section. We discuss some consequences concerning $\operatorname {\mathrm {SL}}^c$-orientations of motivic ring spectra and the étale classifying spaces of certain algebraic groups. In particular, we compute the classifying spaces of diagonalisable groups in the eta-periodic motivic stable homotopy category.

Introduction

Around 40 years ago, Arason computed the Witt groups of projective spaces [Ara80]. This computation was later revisited by Gille [Gil01], Walter [Wal03] and Nenashev [Nen09]. It exhibited Witt groups as a somewhat exotic cohomology theory, whose value on projective spaces differs quite drastically from what is obtained in more classical cohomology theories such as Chow groups or K-theory. It is now understood that this behaviour reflects the lack of $\operatorname {\mathrm {GL}}$ -orientation in Witt theory.

Ananyevskiy observed in [Ana16a] that the key property of Witt groups permitting to perform these computations turns out to be the fact that the Hopf map

$$\begin{align*}\eta \colon {\mathbb{A}}^2 \smallsetminus \{0\} \to \mathbb{P}^1, \quad (x:y) \mapsto [x:y] \end{align*}$$

induces by pullback an isomorphism of Witt groups $\operatorname {\mathrm {W}}(\mathbb {P}^1) \xrightarrow {\sim } \operatorname {\mathrm {W}}({\mathbb {A}}^2 \smallsetminus \{0\})$ . He thus extended in [Ana16a] the above-mentioned computations to arbitrary cohomology theories in which $\eta $ induces an isomorphism.

Inverting the Hopf map $\eta $ in the motivic stable homotopy category $\operatorname {\mathrm {SH}}(S)$ over a base scheme S yields its $\eta $ -periodic version $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , a category which has been studied in details by Bachmann–Hopkins [BH20]. In this paper, we lift Ananyevskiy’s computations of the cohomology of projective spaces to the $\eta $ -periodic stable homotopy category: We obtain for instance that a projective bundle of even relative dimension becomes an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

Another familiar feature of Witt groups is that twisting these groups by squares of line bundles has no effect, which may be viewed as a manifestation of the $\operatorname {\mathrm {SL}}^c$ -orientability of Witt groups (see below). We show that that property of Witt groups in fact follows from their $\eta $ -periodicity alone (see (4.3.2) for a more general statement).

Proposition. Let $V \to X$ be a vector bundle and $L \to X$ a line bundle. Then we have an isomorphism of Thom spaces

$$\begin{align*}\operatorname{\mathrm{Th}}_X(V) \simeq \operatorname{\mathrm{Th}}_X(V \otimes L^{\otimes 2}) \in \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align*}$$

Panin and Walter introduced [PW18, §3] the notion of $\operatorname {\mathrm {SL}}^c$ -orientability for algebraic cohomology theories, which consists of the data of Thom classes for vector bundles equipped with a square root of their determinant and proved that Hermitian K-theory is $\operatorname {\mathrm {SL}}^c$ -oriented. Ananyevskiy later showed [Ana20, Theorem 1.2] that a cohomology theory is $\operatorname {\mathrm {SL}}^c$ -oriented as soon as it is a Zariski sheaf in bidegree $(0,0)$ and pointed out [Ana20, Theorem 1.1] the close relations between $\operatorname {\mathrm {SL}}^c$ -orientations and $\operatorname {\mathrm {SL}}$ -orientations (the latter consisting in the data of Thom classes for vector bundles with trivialised determinant). We show in this paper that the two notions actually coincide in the $\eta $ -periodic context.

Theorem. Every $\operatorname {\mathrm {SL}}$ -orientation of an $\eta $ -periodic motivic commutative ring spectrum is induced by a unique $\operatorname {\mathrm {SL}}^c$ -orientation.

These results are obtained as consequences of the following observation.

Proposition. Let E be a vector bundle of odd rank over a smooth S-scheme X and $E^\circ $ the complement of the zero-section in E.

1. (i) The projection $E^\circ \to X$ admits a section in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

2. (ii) The diagram $E^\circ \times _X E^\circ \rightrightarrows E^\circ \to X$ becomes a split coequaliser diagram in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

The first assertion may be viewed as a splitting principle, while the second permits performing a form of descent. To some extent, this proposition allows us to assume that odd rank vector bundles admit a nowhere-vanishing section (once $\eta $ is inverted); in particular that line bundles are trivial.

Finally, we provide applications to the computation in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ of the étale classifying spaces of certain algebraic groups.

Theorem. For $r \in \mathbb {N} \smallsetminus \{0\}$ , there exist natural maps

$$\begin{align*}S \to \mathrm{B}\mathbb{G}_m \quad ;\quad S \to \mathrm{B} \mu_{2r+1} \quad; \quad \mathbb{G}_m \to \mathrm{B} \mu_{2r} \end{align*}$$

which become isomorphisms in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

From this theorem, we deduce a computation in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ of the classifying space of an arbitrary diagonalisable group. We obtain that all invariants of torsors under a diagonalisable group with values in an $\eta $ -periodic cohomology theory arise from a single invariant of $\mu _2$ -torsors. In the appendix, we present an explicit construction of that invariant, exploiting the identification of the group $\mu _2$ with the orthogonal group $\operatorname {\mathrm {O}}_1$ .

Next, we obtain ‘relative’ computations in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ of certain étale classifying spaces in terms of others.

Theorem. For $n\in \mathbb {N}\smallsetminus \{0\}$ and $r\in \mathbb {N}$ , the natural morphisms

$$\begin{align*}\operatorname{\mathrm{BSL}}_n \to \operatorname{\mathrm{BSL}}^c_n \quad; \quad \operatorname{\mathrm{BGL}}_{2r} \to \operatorname{\mathrm{BGL}}_{2r+1} \quad; \quad \operatorname{\mathrm{BSL}}_{2r+1} \to \operatorname{\mathrm{BGL}}_{2r+1} \end{align*}$$

become isomorphisms in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

The first result can be viewed as a companion of the theorem on orientations stated above and cements the idea that the groups $\operatorname {\mathrm {SL}}_n^c$ and $\operatorname {\mathrm {SL}}_n$ are the same in the eyes of $\eta $ -periodic stable homotopy theory. The second (resp. third) result expresses the fact that odd-dimensional vector bundles behave as if they had a nowhere-vanishing section (resp. trivial determinant) from the point of view of $\eta $ -periodic stable homotopy theory.

The morphism $\operatorname {\mathrm {BSL}}_{2r} \to \operatorname {\mathrm {BGL}}_{2r}$ is not an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , but we show that it admits a section, expressing the fact that every invariant (with values in an $\eta $ -periodic cohomology theory) of even-dimensional vector bundles is determined by its value on those bundles having trivial determinant.

Finally, let us mention that the results of this paper serve as a starting point for the paper [Hau22] on Pontryagin classes.

1 Notation and basic facts

1.1

Throughout the paper, we work over a noetherian base scheme S of finite dimension. The category of smooth separated S-schemes of finite type will be denoted by $\operatorname {\mathrm {Sm}}_S$ . All schemes will be implicitly assumed to belong to $\operatorname {\mathrm {Sm}}_S$ , and the notation ${\mathbb {A}}^n,\mathbb {P}^n,\mathbb {G}_m$ will refer to the corresponding S-schemes. We will denote by $1$ the trivial line bundle over a given scheme in $\operatorname {\mathrm {Sm}}_S$ .

1.2

We will use the ${\mathbb {A}}^1$ -homotopy theory introduced by Morel–Voevodsky [MV99]. We will denote by $\operatorname {\mathrm {Spc}}(S)$ the category of motivic spaces (i.e., simplicial presheaves on $\operatorname {\mathrm {Sm}}_S$ ), by ${\mathrm{Spc}_{\bullet }}(S)$ its pointed version and by $\operatorname {\mathrm {Spt}}(S)$ the category of T-spectra, where ${T= {\mathbb {A}}^1/\mathbb {G}_m}$ . We endow these with the motivic equivalences, resp. stable motivic equivalences, and denote by $\operatorname {\mathrm {H}}(S), \mathrm {H}_{\bullet }(S), \operatorname {\mathrm {SH}}(S)$ the respective homotopy categories. We refer to, for example, [PPR09, Appendix A] for more details.

We have an infinite suspension functor $\Sigma ^{\infty }\colon {\mathrm{Spc}_{\bullet }}(S) \to \operatorname {\mathrm {Spt}}(S)$ . Composing with the functor $\operatorname {\mathrm {Spc}}(S) \to {\mathrm{Spc}_{\bullet }}(S)$ adding an external base point, we obtain a functor $\Sigma ^{\infty }_+ \colon \operatorname {\mathrm {Spc}}(S) \to \operatorname {\mathrm {Spt}}(S)$ .

The spheres are denoted as usual by $S^{p,q} \in {\mathrm{Spc}_{\bullet }}(S)$ for $p,q\in \mathbb {N}$ with $p\geq q$ (where $T\simeq S^{2,1}$ ). The motivic sphere spectrum $\Sigma ^{\infty }_+ S$ will be denoted by $\mathbf {1}_S \in \operatorname {\mathrm {Spt}}(S)$ . When A is a motivic spectrum, we denote its $(p,q)$ -th suspension by $\Sigma ^{p,q}A = S^{p,q} \wedge A$ . This yields functors $\Sigma ^{p,q} \colon \operatorname {\mathrm {SH}}(S) \to \operatorname {\mathrm {SH}}(S)$ for $p,q\in \mathbb {Z}$ .

1.3

When $E\to X$ is a vector bundle with $X \in \operatorname {\mathrm {Sm}}_S$ , we denote by $E^{\circ }=E \smallsetminus X$ the complement of the zero section. The Thom space of E is the pointed motivic space $\operatorname {\mathrm {Th}}_X(E) = E/E^\circ $ . We will write $\operatorname {\mathrm {Th}}_X(E) \in \operatorname {\mathrm {Spt}}(S)$ instead of $\Sigma ^{\infty } \operatorname {\mathrm {Th}}_X(E)$ , in order to lighten the notation. When $g \colon Y \to X$ is a morphism in $\operatorname {\mathrm {Sm}}_S$ , we will usually write $\operatorname {\mathrm {Th}}_Y(E)$ instead of $\operatorname {\mathrm {Th}}_Y(g^*E)$ . Since $E \to X$ is a weak equivalence, we have a cofiber sequence in ${\mathrm{Spc}_{\bullet }}(S)$ , where $p \colon E^\circ \to X$ is the projection,

(1.3.a) $$ \begin{align} (E^\circ)_+ \xrightarrow{p_+} X_+ \to \operatorname{\mathrm{Th}}_X(E). \end{align} $$

If $F \to S$ is a vector bundle and $f \colon X \to S$ the structural morphism, we have by [MV99, Proposition 3.2.17 (1)] a natural identification in ${\mathrm{Spc}_{\bullet }}(S)$

(1.3.b) $$ \begin{align} \operatorname{\mathrm{Th}}_X(E \oplus f^*F) = \operatorname{\mathrm{Th}}_X(E) \wedge \operatorname{\mathrm{Th}}_S(F). \end{align} $$

When $V \to S$ is a vector bundle, we denote by $\Sigma ^V \colon \operatorname {\mathrm {SH}}(S) \to \operatorname {\mathrm {SH}}(S)$ the derived functor induced by $A \mapsto A \wedge \operatorname {\mathrm {Th}}_S(V)$ . It is an equivalence of categories, with inverse denoted by $\Sigma ^{-V}$ .

1.4

Let $i \colon Y\to X$ be a closed immersion in $\operatorname {\mathrm {Sm}}_S$ , with normal bundle $N \to Y$ , and open complement $u\colon U \to X$ . The purity equivalence $X/U \simeq \operatorname {\mathrm {Th}}_Y(N)$ (see, e.g., [MV99, Theorem 3.2.23]) yields a cofiber sequence in ${\mathrm{Spc}_{\bullet }}(S)$

(1.4.a) $$ \begin{align} U_+ \xrightarrow{u_+} X_+ \to \operatorname{\mathrm{Th}}_Y(N). \end{align} $$

More generally, if $V \to X$ is a vector bundle, we have a cofiber sequence in ${\mathrm{Spc}_{\bullet }}(S)$

$$\begin{align*}\operatorname{\mathrm{Th}}_U(V) \to \operatorname{\mathrm{Th}}_X(V) \to \operatorname{\mathrm{Th}}_Y(N \oplus i^*V). \end{align*}$$

This may be deduced from equation (1.4.a) by first reducing to the case $X=S$ using the functor $f_\sharp $ of (1.9) below and then applying the functor $- \wedge \operatorname {\mathrm {Th}}_X(V)$ (both of which preserve homotopy colimits), in view of equation (1.3.b).

1.5

Let $\varphi \colon E \xrightarrow {\sim } F$ be an isomorphism of vector bundles over $X \in \operatorname {\mathrm {Sm}}_S$ . Then $\varphi $ induces a weak equivalence in ${\mathrm{Spc}_{\bullet }}(S)$ (and $\operatorname {\mathrm {Spt}}(S)$ )

$$\begin{align*}\operatorname{\mathrm{Th}}(\varphi) \colon \operatorname{\mathrm{Th}}_X(E) \to \operatorname{\mathrm{Th}}_X(F). \end{align*}$$

If $\psi \colon F \xrightarrow {\sim } G$ is an isomorphism of vector bundles over X, we have

(1.5.a) $$ \begin{align} \operatorname{\mathrm{Th}}(\psi \circ \varphi) = \operatorname{\mathrm{Th}}(\psi) \circ \operatorname{\mathrm{Th}}(\varphi). \end{align} $$

If $X=S$ and $f\in \operatorname {\mathrm {End}}_{\operatorname {\mathrm {SH}}(S)}(\mathbf {1}_S)$ , then we have in $\operatorname {\mathrm {SH}}(S)$

(1.5.b) $$ \begin{align} (\Sigma^Ff) \circ \operatorname{\mathrm{Th}}(\varphi) = \operatorname{\mathrm{Th}}(\varphi) \circ (\Sigma^Ef) \colon \operatorname{\mathrm{Th}}_S(E) \to \operatorname{\mathrm{Th}}_S(F). \end{align} $$

(This follows from the fact that, as morphisms $\mathbf {1}_S \wedge \operatorname {\mathrm {Th}}_S(E) \to \mathbf {1}_S \wedge \operatorname {\mathrm {Th}}_S(F)$

$$\begin{align*}(f \wedge \operatorname{\mathrm{id}}_{\operatorname{\mathrm{Th}}_S(F)}) \circ (\operatorname{\mathrm{id}}_{\mathbf{1}_S} \wedge \operatorname{\mathrm{Th}}(\varphi)) = f \wedge \operatorname{\mathrm{Th}}(\varphi) = (\operatorname{\mathrm{id}}_{\mathbf{1}_S} \wedge \operatorname{\mathrm{Th}}(\varphi)) \circ (f \wedge \operatorname{\mathrm{id}}_{\operatorname{\mathrm{Th}}_S(E)}).) \end{align*}$$

1.6

(See [Mor04, Lemma 6.3.4].) Let $X \in \operatorname {\mathrm {Sm}}_S$ and $u\in H^0(X,\mathbb {G}_m)$ . Consider the automorphism $u \operatorname {\mathrm {id}}_1 \colon 1 \to 1$ of the trivial line bundle over X, and set

$$\begin{align*}\langle u \rangle = \Sigma^{-2,-1} \operatorname{\mathrm{Th}}(u \operatorname{\mathrm{id}}_1) \in \operatorname{\mathrm{Aut}}_{\operatorname{\mathrm{SH}}(S)}(\Sigma^{\infty}_+ X). \end{align*}$$

It follows from equation (1.5.a) that

(1.6.a) $$ \begin{align} \langle uv \rangle = \langle u \rangle \circ \langle v \rangle \quad \text{ for } u,v \in H^0(X,\mathbb{G}_m). \end{align} $$

When $X=S$ and $A \in \operatorname {\mathrm {SH}}(S)$ , we will denote again by $\langle u \rangle \in \operatorname {\mathrm {Aut}}_{\operatorname {\mathrm {SH}}(S)}(A)$ the morphism

$$\begin{align*}A = \mathbf{1}_S \wedge A \xrightarrow{\langle u \rangle \wedge \operatorname{\mathrm{id}}_A} \mathbf{1}_S \wedge A=A. \end{align*}$$

If $f\colon A \to B$ is a morphism in $\operatorname {\mathrm {SH}}(S)$ , then

(1.6.b) $$ \begin{align} f \circ \langle u \rangle = \langle u \rangle \circ f. \end{align} $$

1.7

We denote by $\eta \colon {\mathbb {A}}^2 \smallsetminus \{0\} \to \mathbb {P}^1$ in ${\mathrm{Spc}_{\bullet }}(S)$ the map $(x,y) \mapsto [x:y]$ , where ${\mathbb {A}}^2\smallsetminus \{0\}$ is pointed by $(1,1)$ and $\mathbb {P}^1$ by $[1:1]$ .

1.8

We will consider the categories ${\mathrm{Spc}_{\bullet }}(S)[\eta ^{-1}]$ and $\operatorname {\mathrm {Spt}}(S)[\eta ^{-1}]$ obtained by monoidally inverting the map $\eta $ of equation (1.7), which can be constructed as left Bousfield localisations, as discussed in [Bac18, §6]. Their respective homotopy categories will be denoted by $H_{\bullet }(S)[\eta ^{-1}]$ and $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , and we will usually omit the mention of the localisation functors.

A spectrum $A \in \operatorname {\mathrm {Spt}}(S)$ is called $\eta $ -periodic if the map

(1.8.a) $$ \begin{align} A \wedge \Sigma^{\infty} ({\mathbb{A}}^2 \smallsetminus \{0\}) \xrightarrow{\operatorname{\mathrm{id}} \wedge \Sigma^{\infty} \eta} A \wedge \Sigma^{\infty} \mathbb{P}^1 \end{align} $$

is an isomorphism in $\operatorname {\mathrm {SH}}(S)$ . The full subcategory of such objects in $\operatorname {\mathrm {Spt}}(S)$ can be identified $\operatorname {\mathrm {Spt}}(S)[\eta ^{-1}]$ .

1.9

Let $X \in \operatorname {\mathrm {Sm}}_S$ with structural morphism $f\colon X \to S$ . Then there are Quillen adjunctions

$$\begin{align*}f_\sharp \colon {\mathrm{Spc}_{\bullet}}(X) \leftrightarrows {\mathrm{Spc}_{\bullet}}(S) \colon f^* \quad; \quad f_\sharp \colon \operatorname{\mathrm{Spt}}(X) \leftrightarrows \operatorname{\mathrm{Spt}}(S) \colon f^*. \end{align*}$$

The functor $f^*$ is induced by base-change, while $f_\sharp $ arises from viewing a smooth X-scheme as a smooth S-scheme by composing with f. These induce Quillen adjunctions

$$\begin{align*}f_\sharp \colon {\mathrm{Spc}_{\bullet}}(S)[\eta^{-1}] \leftrightarrows {\mathrm{Spc}_{\bullet}}(S)[\eta^{-1}] \colon f^* \quad; \quad f_\sharp \colon \operatorname{\mathrm{Spt}}(X)[\eta^{-1}] \leftrightarrows \operatorname{\mathrm{Spt}}(S)[\eta^{-1}] \colon f^*. \end{align*}$$

We will also use the notation $f^*,f_{\sharp }$ for the derived functors on the respective homotopy categories.

1.10

(See, e.g., [DHI04].) Let $V \to X$ be a vector bundle with $X \in \operatorname {\mathrm {Sm}}_S$ and $U_\alpha $ an open covering of X. Then the map

$$\begin{align*}\operatorname{\mathrm{hocolim}} \Big( \cdots\mathrel{\substack{\scriptstyle\rightarrow\\[-0.55ex] \scriptstyle\rightarrow \\[-0.55ex] \scriptstyle\rightarrow}} \coprod_{\alpha,\beta} \operatorname{\mathrm{Th}}_{U_\alpha \cap U_\beta}(V|_{U_\alpha \cap U_\beta}) \rightrightarrows \coprod_\alpha \operatorname{\mathrm{Th}}_{U_\alpha}(V|_{U_\alpha})\Big) \to \operatorname{\mathrm{Th}}_X(V) \end{align*}$$

is a weak equivalence.

2 Splitting $\mathbb {G}_m$ -torsors

2.1 Local splitting

In this section, we consider the schemes ${\mathbb {A}}^1,\mathbb {G}_m,\mathbb {P}^1,{\mathbb {A}}^2\smallsetminus \{0\}$ as pointed motivic spaces, respectively via $1,1,[1:1],(1,1)$ . We recall that $T={\mathbb {A}}^1/\mathbb {G}_m$ . We have a chain of weak equivalences

(2.1.0.a) $$ \begin{align} T = {\mathbb{A}}^1/\mathbb{G}_m \xrightarrow{\sim} \mathbb{P}^1/{\mathbb{A}}^1 \xleftarrow{\sim} \mathbb{P}^1, \end{align} $$

where the first arrow is induced by the immersion ${\mathbb {A}}^1 \to \mathbb {P}^1, x \mapsto [x:1]$ , and the quotient $\mathbb {P}^1/{\mathbb {A}}^1$ is taken with respect to the immersion ${\mathbb {A}}^1 \to \mathbb {P}^1, y \mapsto [1:y]$ .

We first recall a well-known fact (see, e.g., [Ana20, Lemma 6.2] for a stable version):

2.1.1 Lemma. Let $u \in H^0(S,\mathbb {G}_m)$ . Then the morphism $\operatorname {\mathrm {Th}}(u^2 \operatorname {\mathrm {id}}_1) \colon T \to T$ (see (1.5)) coincides with the identity in $\mathrm{H}_{\bullet }(S)$ .

Proof. The endomorphism $\varphi \colon \mathbb {P}^1 \to \mathbb {P}^1$ given by $[x:y] \mapsto [u^2x:y]=[ux:u^{-1}y]$ is induced by the matrix

$$\begin{align*}A= \begin{pmatrix} u & 0\\ 0 & u^{-1}\\ \end{pmatrix}. \end{align*}$$

Since

$$\begin{align*}A = \begin{pmatrix} 1 & u\\ 0 & 1\\ \end{pmatrix} \begin{pmatrix} 1 & 0\\ -u^{-1} & 1\\ \end{pmatrix} \begin{pmatrix} 1 & u-1\\ 0 & 1\\ \end{pmatrix} \begin{pmatrix} 1 & 0\\ 1 & 1\\ \end{pmatrix} \begin{pmatrix} 1 & -1\\ 0 & 1\\ \end{pmatrix} \end{align*}$$

is a product of transvections, the endomorphism $\varphi $ induces the identity endomorphism of $\mathbb {P}^1_+$ in $\mathrm{H}_{\bullet }(S)$ (see, e.g., [Ana16a, Lemma 1]). The map $\varphi $ stabilises the copies of ${\mathbb {A}}^1$ given by $x \mapsto [x:1]$ and $y \mapsto [1:y]$ and restricts to $u^2 \operatorname {\mathrm {id}}_1$ on the former. Thus, the statement follows from the isomorphism (2.1.0.a).

2.1.2

Excision yields isomorphisms in $\mathrm{H}_{\bullet }(S)$

$$\begin{align*}({\mathbb{A}}^2 \smallsetminus \{0\})/(\mathbb{G}_m \times {\mathbb{A}}^1) \xleftarrow{\sim} ({\mathbb{A}}^1 \times \mathbb{G}_m)/(\mathbb{G}_m \times \mathbb{G}_m) \xrightarrow{\sim} ({\mathbb{A}}^1/\mathbb{G}_m) \wedge (\mathbb{G}_m)_+ = T \wedge (\mathbb{G}_m)_+. \end{align*}$$

Composing with the quotient ${\mathbb {A}}^2 \smallsetminus \{0\} \to ({\mathbb {A}}^2 \smallsetminus \{0\})/({\mathbb {A}}^1 \times \mathbb {G}_m)$ , this yields a map

(2.1.5.a) $$ \begin{align} {\mathbb{A}}^2 \smallsetminus \{0\} \to T \wedge (\mathbb{G}_m)_+. \end{align} $$

Lemma 2.1.3. The projection $p\colon \mathbb {G}_m \to S$ induces a cofiber sequence in ${\mathrm{Spc}_{\bullet }}(S)$

$$\begin{align*}{\mathbb{A}}^2 \smallsetminus \{0\} \xrightarrow{(2.1.2.a)} T \wedge (\mathbb{G}_m)_+ \xrightarrow{\operatorname{\mathrm{id}} \wedge p_+} T. \end{align*}$$

Proof. This follows from the consideration of the following commutative diagram in ${\mathrm{Spc}_{\bullet }}(S)$ , whose rows are cofiber sequences

and where the curved arrow is the weak equivalence induced by the projection ${\mathbb {G}_m \times {\mathbb {A}}^1 \to \mathbb {G}_m}$ .

The next lemma is reminiscent of [Ana16b, Theorem 3.8]:

Lemma 2.1.4. The morphism $\eta $ of (1.7) factors in $\mathrm{H}_{\bullet }(S)$ as

$$\begin{align*}{\mathbb{A}}^2 \smallsetminus \{0\} \xrightarrow{(2.1.2.a)} T \wedge (\mathbb{G}_m)_+ \xrightarrow{\operatorname{\mathrm{Th}}(t^{-1}\operatorname{\mathrm{id}}_1)} T \wedge (\mathbb{G}_m)_+ \xrightarrow{\operatorname{\mathrm{id}} \wedge p_+} T \xrightarrow{(2.1.0.a)} \mathbb{P}^1, \end{align*}$$

where $t \in H^0(\mathbb {G}_m,\mathbb {G}_m)$ is the tautological section and $p\colon \mathbb {G}_m \to S$ the projection (and $T \wedge (\mathbb {G}_m)_+$ is identified with $\operatorname {\mathrm {Th}}_{\mathbb {G}_m}(1)$ ).

Proof. Consider the commutative diagram in $\operatorname {\mathrm {Sm}}_S$

where the upper horizontal arrow is the natural open immersion, the lower horizontal arrow is given by $x \mapsto [x:1]$ and $\mu $ is given by $(x,y) \mapsto xy^{-1}$ . Excision yields the isomorphisms in the commutative diagram in $\mathrm{H}_{\bullet }(S)$

where $\tilde {\mu }$ is induced by $\mu $ , and the lower horizontal arrows are the morphisms of equation (2.1.0.a). To conclude, observe that the morphism $\tilde {\mu }$ factors as the upper horizontal composite in the following commutative diagram in $H_{\bullet }(S)$ :

Proposition 2.1.5. Let $p \colon \mathbb {G}_m \to S$ be the projection, and $T = {\mathbb {A}}^1/\mathbb {G}_m=\operatorname {\mathrm {Th}}_S(1)$ . Consider the composite (see (1.5))

$$\begin{align*}\pi \colon T \wedge (\mathbb{G}_m)_+ \xrightarrow{\operatorname{\mathrm{Th}}(t \operatorname{\mathrm{id}}_1)} T \wedge (\mathbb{G}_m)_+ \xrightarrow{\operatorname{\mathrm{id}} \wedge p_+} T. \end{align*}$$

Then the following square is homotopy co-Cartesian in ${\mathrm{Spc}_{\bullet }}(S)[\eta ^{-1}]$

Proof. From (2.1.3), we deduce a commutative diagram

(2.1.5.a)

where the left inner square is homotopy co-Cartesian. Applying (2.1.1) over the base $\mathbb {G}_m$ and using the functor $p_\sharp $ of (1.9), we have in $H_{\bullet }(S)$

$$\begin{align*}\operatorname{\mathrm{Th}}(t \operatorname{\mathrm{id}}_1) = \operatorname{\mathrm{Th}}(t^{-1} \operatorname{\mathrm{id}}_1) \colon T \wedge (\mathbb{G}_m)_+ \to T \wedge (\mathbb{G}_m)_+. \end{align*}$$

It thus follows from (2.1.4) that the upper composite in the diagram (2.1.5.a) is an isomorphism in $H_{\bullet }(S)[\eta ^{-1}]$ , hence the exterior square in equation (2.1.5.a) is homotopy co-Cartesian. We conclude that the right inner square is homotopy co-Cartesian (by [Hir03, Proposition 13.3.15, Remark 7.1.10]).

2.2 Global splitting

Definition 2.2.1. Let $L\to S$ be a line bundle. Denote by $L^\circ $ the complement of the zero-section in L, and by $p\colon L^{\circ } \to S$ the projection. Then the graph $L^\circ \to L^\circ \times _S L$ of the open immersion $L^\circ \to L$ may be viewed as a nowhere-vanishing section of the line bundle $p^*L$ over $L^\circ $ , which induces the tautological trivialisation $\tau \colon 1 \xrightarrow {\sim } p^*L$ . We define a morphism in $H_{\bullet }(S)$ (recall that $T=\operatorname {\mathrm {Th}}_S(1)$ , and see (1.5))

$$\begin{align*}\pi_L \colon T \wedge (L^\circ)_+ = \operatorname{\mathrm{Th}}_{L^\circ}(1) \xrightarrow{\operatorname{\mathrm{Th}}(\tau)} \operatorname{\mathrm{Th}}_{L^\circ}(p^*L) \xrightarrow{p} \operatorname{\mathrm{Th}}_S(L). \end{align*}$$

Example 2.2.2. Assume that $L=1$ . Then the tautological trivialisation $\tau \colon 1 \to 1$ of the trivial line bundle over $L^\circ = \mathbb {G}_m$ is given by multiplication by the tautological section $t \in H^0(\mathbb {G}_m,\mathbb {G}_m)$ , hence $\pi _1$ coincides with morphism $\pi $ of (2.1.5).

2.2.3

Let $L\to S$ be a line bundle. If $f \colon R \to S$ is a scheme morphism, then the functor $f^* \colon {\mathrm{Spc}_{\bullet }}(S) \to {\mathrm{Spc}_{\bullet }}(R)$ maps $\pi _L$ to $\pi _{f^*L}$ .

2.2.4

If $\varphi \colon L \xrightarrow {\sim } M$ is an isomorphism of line bundles over S, then the following diagram commutes in $H_{\bullet }(S)$

Proposition 2.2.5. Let $L \to S$ be a line bundle. Then the following square is homotopy co-Cartesian in ${\mathrm{Spc}_{\bullet }}(S)[\eta ^{-1}]$

Proof. Let F be the homotopy colimit of the diagram $\operatorname {\mathrm {Th}}_S(L) \xleftarrow {\pi _L} T \wedge (L^\circ )_+ \xrightarrow {\operatorname {\mathrm {id}} \wedge p_+} T$ . By (1.10) and (1.9) (and in view of (2.2.3)), the fact that $F \simeq *$ may be verified Zariski locally on S. We may thus assume that L is trivial. By (2.2.4), we may further assume that $L=1$ so that $L^\circ = \mathbb {G}_m$ . Then, in view of (2.2.2) the result follows from (2.1.5).

The next statement was initially inspired by [Lev19, Proof of Theorem 4.1].

Corollary 2.2.6. Let $L \to S$ be a line bundle. Then in the notation of (2.2.1), we have an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$

$$\begin{align*}(\Sigma^{\infty}_+ p,\Sigma^{-2,-1}\Sigma^{\infty} \pi_L) \colon \Sigma^{\infty}_+ L^{\circ} \xrightarrow{\sim} \mathbf{1}_S \oplus \Sigma^{-2,-1}\operatorname{\mathrm{Th}}_S(L). \end{align*}$$

Proof. The square induced in $\operatorname {\mathrm {Spt}}(S)[\eta ^{-1}]$ by the square of (2.2.5) is homotopy co-Cartesian, hence also homotopy Cartesian (see, e.g., [Hov99, Remark 7.1.12]). This yields an isomorphism

$$\begin{align*}(\Sigma^{\infty} (\operatorname{\mathrm{id}} \wedge p_+),\Sigma^{\infty} \pi_L) \colon \Sigma^{\infty} (T \wedge (L^{\circ})_+) \xrightarrow{\sim} \Sigma^{\infty} T \oplus \operatorname{\mathrm{Th}}_S(L), \end{align*}$$

from which the result follows by applying the functor $\Sigma ^{-2,-1}$ .

Corollary 2.2.7. Let $L\to S$ be a line bundle and $V \to S$ a vector bundle.

1. (i) The natural map $\operatorname {\mathrm {Th}}_{L^\circ }(V) \to \operatorname {\mathrm {Th}}_S(V)$ extends to a natural isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$

   $$\begin{align*}\operatorname{\mathrm{Th}}_{L^{\circ}}(V) \simeq \operatorname{\mathrm{Th}}_S(V) \oplus \Sigma^{-2,-1}\operatorname{\mathrm{Th}}_S(V \oplus L). \end{align*}$$

2. (ii) Denote by $p \colon L^\circ \to S$ and $p_1,p_2 \colon L^\circ \times _S L^\circ \to L^\circ $ the projections. Then

   $$\begin{align*}\operatorname{\mathrm{Th}}_{L^\circ \times_S L^\circ}(V) \overset{p_1}{\underset{p_2}\rightrightarrows} \operatorname{\mathrm{Th}}_{L^\circ}(V) \xrightarrow{p} \operatorname{\mathrm{Th}}_S(V) \end{align*}$$

   is a split coequaliser diagram in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

Proof. Statement (i) follows by applying the autoequivalence $\Sigma ^V \colon \operatorname {\mathrm {SH}}(S)[\eta ^{-1}] \to \operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ to the decomposition of (2.2.6), in view of equation (1.3.b).

Certainly in the diagram of equation (ii), we have $p \circ p_1 = p \circ p_2$ . The isomorphism (i) yields a section $s \colon \operatorname {\mathrm {Th}}_S(V) \to \operatorname {\mathrm {Th}}_{L^\circ }(V)$ of p in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ . Then, in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , the composite

$$\begin{align*}t \colon \operatorname{\mathrm{Th}}_{L^\circ}(V) = (\Sigma^{\infty}_+ L^\circ) \wedge \operatorname{\mathrm{Th}}_S(V) \xrightarrow{\operatorname{\mathrm{id}} \wedge s} (\Sigma^{\infty}_+ L^\circ) \wedge \operatorname{\mathrm{Th}}_{L^\circ}(V) = \operatorname{\mathrm{Th}}_{L^\circ \times_S L^\circ}(V) \end{align*}$$

is a section of

$$\begin{align*}p_1 \colon \operatorname{\mathrm{Th}}_{L^\circ \times_S L^\circ}(V) = (\Sigma^{\infty}_+ L^\circ) \wedge \operatorname{\mathrm{Th}}_{L^\circ}(V) \xrightarrow{\operatorname{\mathrm{id}} \wedge p} (\Sigma^{\infty}_+ L^\circ) \wedge \operatorname{\mathrm{Th}}_S(V) = \operatorname{\mathrm{Th}}_{L^\circ}(V). \end{align*}$$

On the other hand, in the commutative diagram in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ ,

the upper composite is t, while the lower one is s. Therefore, $p_2 \circ t = s \circ p$ as endomorphisms of $\operatorname {\mathrm {Th}}_{L^\circ }(V)$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , proving equation (ii).

Corollary 2.2.8. Let $L \to S$ be a line bundle, and denote by $p \colon L^\circ \to S$ the projection. Then the functor $p^* \colon \operatorname {\mathrm {SH}}(S)[\eta ^{-1}] \to \operatorname {\mathrm {SH}}(L^\circ )[\eta ^{-1}]$ is faithful and conservative.

Proof. By the smooth projection formula and (2.2.6), the composite $p_\sharp \circ p^* {\colon} \operatorname {\mathrm {SH}}(S)[\eta ^{-1}] \kern1.3pt{\to} \operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ decomposes as

$$\begin{align*}p_\sharp \circ p^* = \operatorname{\mathrm{id}} \wedge (\Sigma^{\infty}_+ L^\circ) = \operatorname{\mathrm{id}} \wedge (\mathbf{1}_S \oplus \Sigma^{-2,-1}\operatorname{\mathrm{Th}}_S(L)) = \operatorname{\mathrm{id}} \oplus (\Sigma^{-2,-1} \circ \Sigma^L), \end{align*}$$

which is faithful, hence so is $p^*$ . The above formula also shows that $p_\sharp \circ p^*$ reflects zero-objects, hence so does $p^*$ . Since $p^*$ is triangulated, it is conservative.

Remark 2.2.9. The results of this section on line bundles will be generalised to odd rank vector bundles in §4.2.

3 Applications to twisted cohomology

3.1 Cohomology theories represented by ring spectra

3.1.1

Let $A \in \operatorname {\mathrm {Spt}}(S)$ be a motivic spectrum. For a pointed motivic space $\mathcal {X}$ , we write

$$\begin{align*}A^{p,q}(\mathcal{X}) = \operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{SH}}(S)}(\Sigma^\infty\mathcal{X},\Sigma^{p,q}A), \end{align*}$$

and $A^{*,*}(\mathcal {X}) = \bigoplus _{p,q\in \mathbb {Z}} A^{p,q}(\mathcal {X})$ . When X is a smooth S-scheme, we will write $A^{*,*}(X)$ instead of $A^{*,*}(X_+)$ . If $E \to X$ is a vector bundle of constant rank r, we write

$$\begin{align*}A^{p,q}(X;E) = A^{p+2r,q+r}(\operatorname{\mathrm{Th}}_X(E)) \end{align*}$$

and extend this notation to arbitrary vector bundles in an obvious way. A morphism $f \colon \mathcal {Y} \to \mathcal {X}$ of pointed motivic spaces (resp. of smooth S-schemes) induces a pullback $f^* \colon A^{*,*}(\mathcal {X}) \to A^{*,*}(\mathcal {Y})$ .

3.1.2

A commutative ring spectrum will mean a commutative monoid in $(\operatorname {\mathrm {SH}}(S),\wedge ,\mathbf {1}_S)$ . When $A \in \operatorname {\mathrm {SH}}(S)$ is a commutative ring spectrum and $X \in \operatorname {\mathrm {Sm}}_S$ , then $A^{*,*}(X)$ is naturally a ring, and $A^{*,*}(X;E)$ an $A^{*,*}(X)$ -module. When $u \in H^0(X,\mathbb {G}_m)$ , we will write $\langle u \rangle \in A^{0,0}(X)$ instead of $\langle u \rangle ^*(1)$ (see (1.6)).

3.1.3

If A is an $\eta $ -periodic motivic spectrum, for any pointed motivic space $\mathcal {X}$ , we have natural isomorphisms for $p,q\in \mathbb {Z}$

$$\begin{align*}A^{p,q}(\mathcal{X}) = \operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{SH}}(S)[\eta^{-1}]}(\Sigma^{\infty} \mathcal{X},\Sigma^{p,q}A). \end{align*}$$

Proposition 3.1.4. Let A be an $\eta $ -periodic motivic spectrum. Let $X \in \operatorname {\mathrm {Sm}}_S$ . Let $L \to X$ be a line bundle and $V \to X$ a vector bundle.

1. (i) Denoting by $p \colon L^\circ \to X$ the projection, we have a split short exact sequence

   $$\begin{align*}0 \to A^{*,*}(X;V) \xrightarrow{p^*} A^{*,*}(L^\circ;V) \to A^{*,*}(X;V \oplus L) \to 0. \end{align*}$$

2. (ii) Denoting by $p_1,p_2 \colon L^\circ \times _X L^\circ \to L^\circ $ the projections, we have an exact sequence

   $$\begin{align*}0 \to A^{*,*}(X;V) \xrightarrow{p^*} A^{*,*}(L^\circ;V) \xrightarrow{p_1^*-p_2^*} A^{*,*}(L^\circ \times_X L^\circ;V). \end{align*}$$

Proof. This follows by applying (2.2.7) over the base X to the image of A under the pullback $\operatorname {\mathrm {Spt}}(S) \to \operatorname {\mathrm {Spt}}(X)$ .

3.2 $\operatorname {\mathrm {SL}}$ - and $\operatorname {\mathrm {SL}}^c$ -orientations

Definition 3.2.1. An $\operatorname {\mathrm {SL}}$ -oriented vector bundle over a scheme X is a pair $(E,\delta )$ , where $E \to X$ is a vector bundle and $\delta \colon 1 \xrightarrow {\sim } \det E$ is an isomorphism of line bundles. We will also say that $\delta $ is an $\operatorname {\mathrm {SL}}$ -orientation of the vector bundle $E \to X$ . An isomorphism of $\operatorname {\mathrm {SL}}$ -oriented vector bundles $(E,\delta ) \xrightarrow {\sim } (F,\epsilon )$ is an isomorphism of vector bundles $\varphi \colon E \xrightarrow {\sim } F$ such that $(\det \varphi ) \circ \delta = \epsilon $ .

Definition 3.2.2 See [PW18, §3].

An $\operatorname {\mathrm {SL}}^c$ -oriented vector bundle over a scheme X is a triple $(E,L,\lambda )$ , where $E \to X$ is a vector bundle and $L \to X$ a line bundle, and $\lambda \colon L^{\otimes 2} \xrightarrow {\sim } \det E$ is an isomorphism. We will also say that $(L,\lambda )$ is an $\operatorname {\mathrm {SL}}^c$ -orientation of the vector bundle $E \to X$ . An isomorphism of $\operatorname {\mathrm {SL}}^c$ -oriented vector bundles $(E,L,\lambda ) \xrightarrow {\sim } (F,M,\mu )$ is an isomorphism of vector bundles $\varphi \colon E \xrightarrow {\sim } F$ and an isomorphism of line bundles $\psi \colon L \xrightarrow {\sim } M$ such that $(\det \varphi ) \circ \lambda = \mu \circ \psi ^{\otimes 2}$ .

3.2.3

Observe that each $\operatorname {\mathrm {SL}}$ -orientation $\delta $ of a vector bundle E induces an $\operatorname {\mathrm {SL}}^c$ -orientation $(L,\lambda )$ of E, where $L=1$ and $\lambda $ is the composite $1^{\otimes 2} \simeq 1 \xrightarrow {\delta } \det E$ .

3.2.4

Let $(E,L,\lambda )$ be an $\operatorname {\mathrm {SL}}^c$ -oriented vector bundle, and assume that the line bundle L is trivial. Then every trivialisation $\alpha \colon 1 \xrightarrow {\sim } L$ induces an $\operatorname {\mathrm {SL}}$ -orientation of E given by

$$\begin{align*}\delta_\alpha \colon 1 \simeq 1^{\otimes 2} \xrightarrow{\alpha^{\otimes 2}} L^{\otimes 2} \xrightarrow{\lambda} \det E. \end{align*}$$

Observe that the $\operatorname {\mathrm {SL}}^c$ -oriented vector bundle induced (in the sense of (3.2.3)) by $\delta _\alpha $ is isomorphic to $(E,L,\lambda )$ .

3.2.5

Consider a commutative ring spectrum $A \in \operatorname {\mathrm {SH}}(S)$ . By a $\operatorname {\mathrm {SL}}$ -, resp. $\operatorname {\mathrm {SL}}^c$ -, orientation of A, we will mean a normalised orientation in the sense of [Ana20, Definition 3.3]. Such data consists in Thom classes $\operatorname {\mathrm {th}}_{(E,\delta )} \in A^{*,*}(X;E)$ for each $\operatorname {\mathrm {SL}}$ -oriented vector bundle $(E,\delta )$ over $X \in \operatorname {\mathrm {Sm}}_S$ , resp. $\operatorname {\mathrm {th}}_{(E,L,\lambda )} \in A^{*,*}(X;E)$ for each $\operatorname {\mathrm {SL}}^c$ -oriented vector bundle $(E,L,\lambda )$ over $X \in \operatorname {\mathrm {Sm}}_S$ , satisfying a series of axioms.

3.2.6

Let $A \in \operatorname {\mathrm {SH}}(S)$ be a commutative ring spectrum. Then each $\operatorname {\mathrm {SL}}^c$ -orientation of A induces an $\operatorname {\mathrm {SL}}$ -orientation of A, by letting the Thom class of an $\operatorname {\mathrm {SL}}$ -oriented vector bundle be the Thom class of the induced $\operatorname {\mathrm {SL}}^c$ -oriented vector bundle, in the sense of (3.2.3).

Lemma 3.2.7. Let $A \in \operatorname {\mathrm {SH}}(S)$ be an $\operatorname {\mathrm {SL}}$ -oriented commutative ring spectrum. Let $(E,L,\lambda )$ be an $\operatorname {\mathrm {SL}}^c$ -oriented vector bundle over $X \in \operatorname {\mathrm {Sm}}_S$ , and assume that the line bundle L is trivial. Then, in the notation of (3.2.4), the Thom class $\operatorname {\mathrm {th}}_{(E,\delta _\alpha )} \in A^{*,*}(X;E)$ does not depend on the choice of the trivialisation $\alpha $ of L.

Proof. If $\alpha \colon 1 \xrightarrow {\sim } L$ is a trivialisation, then every trivialisation is of the form $u \alpha $ for some $u \in H^0(X,\mathbb {G}_m)$ . In the notation of (3.2.4), we then have $\delta _{u \alpha } = u^2 \delta _{\alpha }$ . By [Ana20, Lemma 7.3], we have

$$\begin{align*}\operatorname{\mathrm{th}}_{(E,\delta_{u \alpha})} = \operatorname{\mathrm{th}}_{(E,u^2 \delta_{\alpha})} = \langle u^2 \rangle \operatorname{\mathrm{th}}_{(E,\delta_\alpha)} \in A^{*,*}(X;E). \end{align*}$$

Since $\langle u^2 \rangle =1 \in A^{0,0}(X)$ by (2.1.1) (or [Ana20, Lemma 6.2]), the statement follows.

Proposition 3.2.8. Let $A \in \operatorname {\mathrm {SH}}(S)$ be an $\eta $ -periodic commutative ring spectrum. Then every $\operatorname {\mathrm {SL}}$ -orientation of A is induced (in the sense of (3.2.6)) by a unique $\operatorname {\mathrm {SL}}^c$ -orientation.

Proof. We assume given an $\operatorname {\mathrm {SL}}$ -orientation of A. Let $(E,L,\lambda )$ be an $\operatorname {\mathrm {SL}}^c$ -oriented vector bundle. Denoting by $p\colon L^\circ \to X$ the projection, the line bundle $p^*L$ over $L^\circ $ admits a tautological trivialisation $\tau \colon 1 \xrightarrow {\sim } p^*L$ . In view of (3.2.4), this yields an $\operatorname {\mathrm {SL}}$ -orientation $\delta _\tau $ of $p^*E$ .

First, assume given an $\operatorname {\mathrm {SL}}^c$ -orientation of A compatible with its $\operatorname {\mathrm {SL}}$ -orientation. As observed in (3.2.4), the $\operatorname {\mathrm {SL}}^c$ -oriented vector bundle $(p^*E,p^*L,p^*\lambda )$ is isomorphic to the one induced by the $\operatorname {\mathrm {SL}}$ -oriented vector bundle $(p^*E,\delta _\tau )$ . Thus, we must have

$$\begin{align*}p^* \operatorname{\mathrm{th}}_{(E,L,\lambda)} = \operatorname{\mathrm{th}}_{(p^*E,p^*L,p^*\lambda)} = \operatorname{\mathrm{th}}_{(p^*E,\delta_\tau)} \in A^{*,*}(L^{\circ};p^*E). \end{align*}$$

Since $p^* \colon A^{*,*}(X;E) \to A^{*,*}(L^{\circ };p^*E)$ is injective by (3.1.4), we obtain the uniqueness part of the statement.

We now construct an $\operatorname {\mathrm {SL}}^c$ -orientation of A from its $\operatorname {\mathrm {SL}}$ -orientation. In the situation considered at the beginning of the proof, let $p_1,p_2 \colon L^\circ \times _X L^\circ \to L^\circ $ be the projections, and set $q=p \circ p_1=p \circ p_2$ . The tautological trivialisation $\tau $ of $p^*L$ yields two trivialisations $p_1^*\tau $ and $p_2^*\tau $ of $q^*L$ , and thus two $\operatorname {\mathrm {SL}}$ -orientations $\alpha _1=\delta _{p_1^*\tau }$ and $\alpha _2=\delta _{p_2^*\tau }$ of E. However, it follows from (3.2.7) that their Thom classes coincide so that (observe that $\alpha _i = p_i^*(\delta _\tau )$ for $i=1,2$ )

$$\begin{align*}p_1^*\operatorname{\mathrm{th}}_{(p^*E,\delta_\tau)} = \operatorname{\mathrm{th}}_{(E,\alpha_1)} = \operatorname{\mathrm{th}}_{(E,\alpha_2)} = p_2^*\operatorname{\mathrm{th}}_{(p^*E,\delta_\tau)} \in A^{*,*}(L^{\circ} \times_X L^{\circ};q^*E). \end{align*}$$

Therefore, it follows from (3.1.4.ii) that the element $\operatorname {\mathrm {th}}_{(p^*E,\delta _\tau )} \in A^{*,*}(L^\circ ;E)$ is the image of a unique element $\theta _{(E,L,\lambda )} \in A^{*,*}(X;E)$ .

From the fact that $(E,\delta ) \mapsto \operatorname {\mathrm {th}}_{(E,\delta )}$ defines an $\operatorname {\mathrm {SL}}$ -orientation of A, we deduce at once that $(E,L,\lambda ) \mapsto \theta _{(E,L,\lambda )}$ defines an $\operatorname {\mathrm {SL}}^c$ -orientation of A: Indeed, each axiom of [Ana20, Definition 3.3] can be verified after pulling back along $p\colon L^\circ \to X$ since $p^* \colon A^{*,*}(X;E) \to A^{*,*}(L^\circ ;p^*E)$ is injective by (3.1.4).

To conclude the proof, it remains to show the $\operatorname {\mathrm {SL}}^c$ -orientation $(E,L,\lambda ) \mapsto \theta _{(E,L,\lambda )}$ induces the original $\operatorname {\mathrm {SL}}$ -orientation of A. So let us assume that the $\operatorname {\mathrm {SL}}^c$ -oriented vector bundle $(E,L,\lambda )$ is induced by an $\operatorname {\mathrm {SL}}$ -oriented vector bundle $(E,\delta )$ , in the sense of (3.2.3). In particular, $L=1$ . Then the tautological trivialisation $\tau \colon 1 \xrightarrow {\sim } p^*L$ and the trivialisation $1 =p^*1 =p^*L$ yield two $\operatorname {\mathrm {SL}}$ -orientations of $p^*E$ . Their Thom classes in $A^{*,*}(L^\circ ;p^*E)$ coincide by (3.2.7), and they are, respectively, $p^*\theta _{(E,L,\lambda )}$ and $p^*\operatorname {\mathrm {th}}_{(E,\delta )}$ . Since $p^* \colon A^{*,*}(X;E) \to A^{*,*}(L^\circ ;p^*E)$ is injective by (3.1.4), we have $\theta _{(E,L,\lambda )} = \operatorname {\mathrm {th}}_{(E,\delta )} \in A^{*,*}(X;E)$ , as required.

Remark 3.2.9. Ananyevskiy constructed ‘Thom isomorphisms’ associated with $\operatorname {\mathrm {SL}}^c$ -bundles in [Ana20, §4] when A is an arbitrary $\operatorname {\mathrm {SL}}$ -oriented theory, but as explained in [Ana20, Remark 4.4] it is not clear whether this yields an $\operatorname {\mathrm {SL}}^c$ -orientation, the problem being the multiplicativity axiom. When A is $\eta $ -periodic, our construction leads to the same Thom isomorphisms for $\operatorname {\mathrm {SL}}^c$ -bundles (in fact, the proof of (3.2.8) shows that there is at most one way to construct such functorial isomorphisms compatibly with the $\operatorname {\mathrm {SL}}$ -orientation). Thus, the Thom isomorphisms constructed by Ananyevskiy do give rise to an $\operatorname {\mathrm {SL}}^c$ -orientation when A is $\eta $ -periodic.

3.3 Twisting by doubles and squares of line bundles

Proposition 3.3.1. Let L be a line bundle over S.

1. (i) There exist an isomorphism $\Sigma ^{4,2}\mathbf {1}_S \simeq \operatorname {\mathrm {Th}}_S(L^{\oplus 2})$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

2. (ii) If $s_1,s_2 \in \mathbb {Z}$ are of the same parity, there exist an isomorphism

   $$\begin{align*}\operatorname{\mathrm{Th}}_S(L^{\otimes s_1}) \simeq \operatorname{\mathrm{Th}}_S(L^{\otimes s_2}) \; \text{in } \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align*}$$

Proof. Let us first assume that the line bundle $L \to S$ admits a trivialisation $\alpha \colon 1 \xrightarrow {\sim } L$ . Then we have an isomorphism in $\operatorname {\mathrm {SH}}(S)$ (see (1.5))

(3.3.1.a) $$ \begin{align} \operatorname{\mathrm{Th}}(\alpha^{\oplus 2}) \colon \operatorname{\mathrm{Th}}_S(1^{\oplus 2}) \xrightarrow{\sim} \operatorname{\mathrm{Th}}_S(L^{\oplus 2}). \end{align} $$

Every trivialisation of L is of the form $u \alpha $ with $u \in H^0(S,\mathbb {G}_m)$ . As automorphisms of $\operatorname {\mathrm {Th}}_S(1^{\oplus 2})$ in $\operatorname {\mathrm {SH}}(S)$ we have,

$$\begin{align*}\operatorname{\mathrm{Th}}((u\operatorname{\mathrm{id}}_1)^{\oplus 2}) = \operatorname{\mathrm{Th}}((u^2\operatorname{\mathrm{id}}_1) \oplus \operatorname{\mathrm{id}}_1) \circ \operatorname{\mathrm{Th}}((u^{-1}\operatorname{\mathrm{id}}_1) \oplus (u\operatorname{\mathrm{id}}_1)) = \operatorname{\mathrm{Th}}((u^2\operatorname{\mathrm{id}}_1) \oplus \operatorname{\mathrm{id}}_1) \end{align*}$$

because $\operatorname {\mathrm {Th}}((u^{-1}\operatorname {\mathrm {id}}_1) \oplus (u\operatorname {\mathrm {id}}_1))$ is the identity of $\operatorname {\mathrm {Th}}_S(1^{\oplus 2})$ , being given by a product of transvections (see the proof of (2.1.1)). Now, by (2.1.1), under the identification $\operatorname {\mathrm {Th}}_S(1^{\oplus 2})=\operatorname {\mathrm {Th}}_S(1) \wedge \operatorname {\mathrm {Th}}_S(1)$ (see equation (1.3.b)), we have

$$\begin{align*}\operatorname{\mathrm{Th}}((u^2\operatorname{\mathrm{id}}_1) \oplus \operatorname{\mathrm{id}}_1) = \operatorname{\mathrm{Th}}(u^2\operatorname{\mathrm{id}}_1) \wedge \operatorname{\mathrm{id}}_{\operatorname{\mathrm{Th}}_S(1)} = \operatorname{\mathrm{id}}_{\operatorname{\mathrm{Th}}_S(1)} \wedge \operatorname{\mathrm{id}}_{\operatorname{\mathrm{Th}}_S(1)} = \operatorname{\mathrm{id}}_{\operatorname{\mathrm{Th}}_S(1^{\oplus 2})}. \end{align*}$$

Therefore, $\operatorname {\mathrm {Th}}((u\operatorname {\mathrm {id}}_1)^{\oplus 2})$ is the identity of $\operatorname {\mathrm {Th}}_S(1^{\oplus 2})$ in $\operatorname {\mathrm {SH}}(S)$ , hence

$$\begin{align*}\operatorname{\mathrm{Th}}((u\alpha)^{\oplus 2}) = \operatorname{\mathrm{Th}}(\alpha^{\oplus 2}) \circ \operatorname{\mathrm{Th}}((u\operatorname{\mathrm{id}}_1)^{\oplus 2}) = \operatorname{\mathrm{Th}}(\alpha^{\oplus 2}) \end{align*}$$

so that the isomorphism $\operatorname {\mathrm {Th}}(\alpha ^{\oplus 2})$ in $\operatorname {\mathrm {SH}}(S)$ considered in equation (3.3.1.a) is independent of the choice of the trivialisation $\alpha $ .

Next, let us consider the case (ii). If $\alpha \colon 1 \xrightarrow {\sim } L$ is a trivialisation, we have an isomorphism in $\operatorname {\mathrm {SH}}(S)$

(3.3.1.b) $$ \begin{align} \operatorname{\mathrm{Th}}(\operatorname{\mathrm{id}}_{L^{\otimes s_1}} \otimes \alpha^{\otimes s_2-s_1}) \colon \operatorname{\mathrm{Th}}_S(L^{\otimes s_1}) \xrightarrow{\sim} \operatorname{\mathrm{Th}}_S(L^{\otimes s_2}). \end{align} $$

(Here and below, for $r \in \mathbb {N}$ , the notation $\alpha ^{\otimes -r}$ refers to the morphism $((\alpha ^\vee )^{-1})^{\otimes r}$ .) Now for $u \in H^0(S,\mathbb {G}_m)$ , the composite in $\operatorname {\mathrm {SH}}(S)$

$$\begin{align*}\operatorname{\mathrm{Th}}_S(1) \xrightarrow{\operatorname{\mathrm{Th}}(\alpha^{\otimes s_1})} \operatorname{\mathrm{Th}}_S(L^{\otimes s_1}) \xrightarrow{\operatorname{\mathrm{Th}}(\operatorname{\mathrm{id}}_{L^{\otimes s_1}} \otimes (u\alpha)^{\otimes s_2-s_1})} \operatorname{\mathrm{Th}}_S(L^{\otimes s_2}) \xrightarrow{\operatorname{\mathrm{Th}}(\alpha^{\otimes s_2})^{-1}} \operatorname{\mathrm{Th}}_S(1) \end{align*}$$

coincides with $\operatorname {\mathrm {Th}}(u^{s_2-s_1} \operatorname {\mathrm {id}}_1)$ , which is the identity by (2.1.1) (recall that $s_2-s_1$ is even), and in particular does not depend on $u\in H^0(S,\mathbb {G}_m)$ . Since the left and right arrows in the above composite are isomorphisms, we deduce that the middle arrow does not depend on $u\in H^0(S,\mathbb {G}_m)$ , which shows as above that the isomorphism (3.3.1.b) is independent of the choice of the trivialisation $\alpha $ .

Let us come back to the general case, where $L \to S$ is a possibly nontrivial line bundle. Let $p\colon L^\circ \to S$ be the projection, and consider the tautological trivialisation $\tau $ of the line bundle $p^*L$ over $L^\circ $ . Let us consider the isomorphism $\varphi \colon \operatorname {\mathrm {Th}}_{L^\circ }(B) \xrightarrow {\sim } \operatorname {\mathrm {Th}}_{L^\circ }(C)$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , where

• • $B = 1^{\oplus 2}, C=L^{\oplus 2}, \varphi = \operatorname {\mathrm {Th}}(\tau ^{\oplus 2})$ in case (i).

• • $B = L^{\otimes s_1}, C=L^{\otimes s_2}, \varphi = \operatorname {\mathrm {Th}}(\operatorname {\mathrm {id}}_{L^{\otimes s_1}} \otimes \alpha ^{\otimes s_2-s_1})$ in case (ii).

Let $p_1,p_2 \colon L^\circ \times _S L^\circ \to L^\circ $ be the projections, and set $q=p \circ p_1=p \circ p_2$ . For $i\in \{1,2\}$ , the isomorphism $p_i^*\varphi \colon \operatorname {\mathrm {Th}}_{L^\circ \times _S L^\circ }(B) \to \operatorname {\mathrm {Th}}_{L^\circ \times _S L^\circ }(C)$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ is induced by the trivialisation $p_i^*\tau $ of the line bundle $q^*L$ over $L^\circ \times _S L^\circ $ , hence does not depend on i, by the special case considered at the beginning of the proof. Thus, by (2.2.7.ii) there exists a unique morphism f fitting into the commutative diagram in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$

as well as a unique morphism g into the commutative diagram in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$

As $p \colon \operatorname {\mathrm {Th}}_{L^{\circ }}(B) \to \operatorname {\mathrm {Th}}_S(B)$ and $p \colon \operatorname {\mathrm {Th}}_{L^{\circ }}(C) \to \operatorname {\mathrm {Th}}_S(C)$ are epimorphisms in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ (see (2.2.7)), it follows that f and g are mutually inverse isomorphisms in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

Remark 3.3.2. Proposition (3.3.1) will be improved in (4.3.2).

4 Nowhere-vanishing sections of odd rank bundles

4.1 Projective bundles

The results of this section are slight generalisations of those of [Ana16a, §4].

4.1.1

Let us consider the linear embeddings $i_k \colon \mathbb {P}^k \to \mathbb {P}^{k+1}$ given by the vanishing of the last coordinate. Denote by $\iota _k \colon S \to \mathbb {P}^k$ the S-point given by the composite ${S=\mathbb {P}^0 \xrightarrow {i_0} \mathbb {P}^1 \xrightarrow {i_1} \cdots \xrightarrow {i_{k-1}} \mathbb {P}^k}$ .

4.1.2

Assume given a collection $\mathcal {D}=(d_1,\ldots ,d_r) \in \mathbb {Z}^r$ for some $r\in \mathbb {N}$ . We will denote by $\mathcal {O}(\mathcal {D})$ the vector bundle $\mathcal {O}(d_1) \oplus \cdots \oplus \mathcal {O}(d_r)$ over $\mathbb {P}^k$ , for each $k \in \mathbb {N}$ . When $k=0$ , we have a canonical isomorphism $\mathcal {O}(\mathcal {D}) \simeq 1^{\oplus r}$ over $\mathbb {P}^0=S$ . This yields, for any $k\in \mathbb {N}$ , a canonical map in $\operatorname {\mathrm {Spt}}(S)$

(4.1.2.a) $$ \begin{align} \Sigma^{2r,r} \mathbf{1}_S = \operatorname{\mathrm{Th}}_S(1^{\oplus r}) \simeq \operatorname{\mathrm{Th}}_{\mathbb{P}^0}(\mathcal{O}(\mathcal{D})) \xrightarrow{\iota_k} \operatorname{\mathrm{Th}}_{\mathbb{P}^k}(\mathcal{O}(\mathcal{D})). \end{align} $$

Proposition 4.1.3. Let $k,r \in \mathbb {N}$ , and $d_1,\ldots ,d_r \in \mathbb {Z}$ . Set $\mathcal {D}=(d_1,\ldots ,d_r)$ , and let ${d=d_1 +\cdots +d_r}$ . We use the notation $\mathcal {O}(\mathcal {D})$ described in (4.1.2).

1. (i) If k and d are odd, then $\operatorname {\mathrm {Th}}_{\mathbb {P}^k}(\mathcal {O}(\mathcal {D})) = 0$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

2. (ii) If k and d are even, then equation (4.1.2.a) induces an isomorphism $\Sigma ^{2r,r}\mathbf {1}_S \simeq \operatorname {\mathrm {Th}}_{\mathbb {P}^k}(\mathcal {O}(\mathcal {D}))$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

3. (iii) If k is even and d is odd, then $\operatorname {\mathrm {Th}}_{\mathbb {P}^k}(\mathcal {O}(\mathcal {D})) \simeq \Sigma ^{2(k+r),k+r}\mathbf {1}_S$ .

4. (iv) If k is odd and d is even, then $\operatorname {\mathrm {Th}}_{\mathbb {P}^k}(\mathcal {O}(\mathcal {D})) \simeq \Sigma ^{2(k+r),k+r}\mathbf {1}_S \oplus \Sigma ^{2r,r}\mathbf {1}_S$ .

Proof. Let us first prove (i). Assume that k and d are odd. Consider a linear embedding $\mathbb {P}^1 \to \mathbb {P}^k$ . Its normal bundle is $\mathcal {O}(1)^{\oplus k-1}$ , and its open complement is a vector bundle over $\mathbb {P}^{k-2}$ . The corresponding zero-section $\mathbb {P}^{k-2} \to \mathbb {P}^k\smallsetminus \mathbb {P}^1$ induces an isomorphism in $\operatorname {\mathrm {SH}}(S)$ and is the restriction of a linear embedding $\mathbb {P}^{k-2} \to \mathbb {P}^k$ . Thus, (1.4) yields a distinguished triangle in $\operatorname {\mathrm {SH}}(S)$

$$\begin{align*}\operatorname{\mathrm{Th}}_{\mathbb{P}^{k-2}}(\mathcal{O}(\mathcal{D})) \to \operatorname{\mathrm{Th}}_{\mathbb{P}^k}(\mathcal{O}(\mathcal{D})) \to \operatorname{\mathrm{Th}}_{\mathbb{P}^1}(\mathcal{O}(\mathcal{D}) \oplus \mathcal{O}(1)^{\oplus k-1}) \to \Sigma^{1,0} \operatorname{\mathrm{Th}}_{\mathbb{P}^{k-2}}(\mathcal{O}(\mathcal{D})). \end{align*}$$

Using induction on the odd integer k, we are reduced to assuming that $k=1$ . Now, by (3.3.1.ii) we have in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$

$$\begin{align*}\operatorname{\mathrm{Th}}_{\mathbb{P}^1}(\mathcal{O}(\mathcal{D})) \simeq \Sigma^{2s,s}(\operatorname{\mathrm{Th}}_{\mathbb{P}^1}(\mathcal{O}(-1))^{\wedge r-s}), \end{align*}$$

where s is the number of indices $i \in \{1,\dots ,r\}$ such that $d_i$ is even. Since d is odd, so is $r-s$ and using (3.3.1.i), we deduce that

$$\begin{align*}\operatorname{\mathrm{Th}}_{\mathbb{P}^1}(\mathcal{O}(\mathcal{D})) \simeq \Sigma^{2(r-1),r-1}\operatorname{\mathrm{Th}}_{\mathbb{P}^1}(\mathcal{O}(-1)) \quad \text{in } \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align*}$$

But $\operatorname {\mathrm {Th}}_{\mathbb {P}^1}(\mathcal {O}(-1))$ vanishes in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ because of the distinguished triangle (see (1.3.a))

$$\begin{align*}\Sigma^{\infty}_+ \mathcal{O}(-1)^{\circ} \to \Sigma^{\infty}_+ \mathbb{P}^1 \to \operatorname{\mathrm{Th}}_{\mathbb{P}^1}(\mathcal{O}(-1)) \to \Sigma^{1,0}\Sigma^{\infty}_+ \mathcal{O}(-1)^{\circ} \end{align*}$$

and the definition of the map $\eta $ (recall that $\mathcal {O}(-1)^\circ = {\mathbb {A}}^2 \smallsetminus \{0\}$ ). We have proved equation (i).

Let us come back to the situation when k and d are arbitrary. Consider a linear embedding $\mathbb {P}^{k-1} \to \mathbb {P}^k$ avoiding the S-point $\iota _k \colon S \to \mathbb {P}^k$ (we write $\mathbb {P}^{-1}=\varnothing $ ). It is a closed immersion defined by the vanishing of a regular section of $\mathcal {O}(1)$ . Its open complement is isomorphic to ${\mathbb {A}}^k$ , and the morphism $j_k\colon S \to {\mathbb {A}}^k$ induced by $\iota _k$ induces an isomorphism in $\operatorname {\mathrm {SH}}(S)$ . The canonical trivialisation of $\mathcal {O}(\mathcal {D})$ over $\mathbb {P}^0=S$ is the restriction along $j_k$ of a trivialisation of $\mathcal {O}(\mathcal {D})|_{{\mathbb {A}}^k}$ (induced by the trivialisation of $\mathcal {O}(1)|_{{\mathbb {A}}^k}$ corresponding to the regular section of $\mathcal {O}(1)$ mentioned above). It follows that the map $\operatorname {\mathrm {Th}}_{\mathbb {P}^0}(\mathcal {O}(\mathcal {D})) \to \operatorname {\mathrm {Th}}_{{\mathbb {A}}^k}(\mathcal {O}(\mathcal {D}))$ induced by $j_k$ induces an isomorphism in $\operatorname {\mathrm {SH}}(S)$ . Thus, (1.4) yields a distinguished triangle in $\operatorname {\mathrm {SH}}(S)$

$$\begin{align*}\Sigma^{2r,r}\mathbf{1}_S \xrightarrow{(4.1.2.\mathrm{a})} \operatorname{\mathrm{Th}}_{\mathbb{P}^k}(\mathcal{O}(\mathcal{D})) \to \operatorname{\mathrm{Th}}_{\mathbb{P}^{k-1}}(\mathcal{O}(\mathcal{D}) \oplus \mathcal{O}(1)) \to \Sigma^{2r+1,r} \mathbf{1}_S \end{align*}$$

so that equation (ii) follows from equation (i).

Consider now a linear embedding $s\colon S=\mathbb {P}^0 \to \mathbb {P}^k$ avoiding $i_{k-1}(\mathbb {P}^{k-1})$ . Its open complement U is a line bundle over $\mathbb {P}^{k-1}$ . The corresponding zero-section $\mathbb {P}^{k-1} \to U$ induces an isomorphism in $\operatorname {\mathrm {SH}}(S)$ and is the restriction of the linear embedding $i_{k-1} \colon \mathbb {P}^{k-1} \to \mathbb {P}^k$ . Since the vector bundle $s^*\mathcal {O}(\mathcal {D})$ and the normal bundle $s^*\mathcal {O}(1)^{\oplus k}$ to s are both trivial, we have by (1.4) a distinguished triangle in $\operatorname {\mathrm {SH}}(S)$

(4.1.3.a) $$ \begin{align} \operatorname{\mathrm{Th}}_{\mathbb{P}^{k-1}}(\mathcal{O}(\mathcal{D})) \to \operatorname{\mathrm{Th}}_{\mathbb{P}^k}(\mathcal{O}(\mathcal{D})) \to \Sigma^{2(k+r),k+r}\mathbf{1}_S \to \Sigma^{1,0} \operatorname{\mathrm{Th}}_{\mathbb{P}^{k-1}}(\mathcal{O}(\mathcal{D})). \end{align} $$

Therefore, equation (iii) follows from equation (i).

Finally, assume that k is odd and d is even. It follows from equation (ii) that the composite $\operatorname {\mathrm {Th}}_{\mathbb {P}^{k-1}}(\mathcal {O}(\mathcal {D})) \to \operatorname {\mathrm {Th}}_{\mathbb {P}^k}(\mathcal {O}(\mathcal {D})) \to \operatorname {\mathrm {Th}}_{\mathbb {P}^{k+1}}(\mathcal {O}(\mathcal {D}))$ is an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , hence $\operatorname {\mathrm {Th}}_{\mathbb {P}^{k-1}}(\mathcal {O}(\mathcal {D})) \to \operatorname {\mathrm {Th}}_{\mathbb {P}^k}(\mathcal {O}(\mathcal {D}))$ admits a retraction, giving a splitting of the triangle (4.1.3.a). In view of equation (ii), this proves equation (iv).

Corollary 4.1.4. If $k \in \mathbb {N}$ is even, the structural morphism $\mathbb {P}^k \to S$ induces an isomorphism $\Sigma ^{\infty }_+ \mathbb {P}^k \xrightarrow {\sim } \mathbf {1}_S$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

Proof. The structural morphism is retraction of $\iota _k$ , so the corollary follows from (4.1.3.ii) applied with $r=0$ .

Proposition 4.1.5. Let $E,V_1,\ldots ,V_n$ be vector bundles of constant rank over S, and $d_1,\ldots ,d_n \in \mathbb {Z}$ . Assume that $\operatorname {\mathrm {rank}} E$ is even and that $d_1 \operatorname {\mathrm {rank}} V_1 +\cdots +d_n \operatorname {\mathrm {rank}} V_n$ is odd. Then

$$\begin{align*}\operatorname{\mathrm{Th}}_{\mathbb{P}(E)}((\mathcal{O}(d_1)\otimes q^*V_1) \oplus \cdots \oplus (\mathcal{O}(d_n)\otimes q^* V_n))=0 \in \operatorname{\mathrm{SH}}(S)[\eta^{-1}], \end{align*}$$

where $q \colon \mathbb {P}(E) \to S$ is the projective bundle.

Proof. By (1.10) and (1.9), this may be verified Zariski locally on S, so we may assume that $E,V_1,\ldots ,V_n$ are all trivial. Then the statement follows from (4.1.3.i).

Proposition 4.1.6. Let $E,V$ be vector bundles over S. Assume that E has constant odd rank. Then $\operatorname {\mathrm {Th}}_{\mathbb {P}(E)}(V) \to \operatorname {\mathrm {Th}}_S(V)$ is an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ . In particular, $\Sigma ^{\infty }_+ \mathbb {P}(E) \xrightarrow {\sim } \mathbf {1}_S$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

Proof. By (1.10) and (1.9), this may be verified Zariski locally on S, so we may assume that E and V are both trivial. Then the statement follows after suspending (4.1.4).

4.2 Odd rank vector bundles

4.2.1

Let $E\to S$ be a vector bundle. The composite $\mathcal {O}(-1) \subset E \times _S \mathbb {P}(E) \to E$ restricts to an isomorphism $\mathcal {O}(-1)^{\circ } \xrightarrow {\sim } E^\circ $ , which is $\mathbb {G}_m$ -equivariant. We thus obtain a commutative diagram in $\operatorname {\mathrm {Spt}}(S)$

(4.2.1.a)

which induces a morphism of the homotopy cofibers of the horizontal morphisms:

(4.2.1.b) $$ \begin{align} \operatorname{\mathrm{Th}}_{\mathbb{P}(E)}(\mathcal{O}(-1)) \to \operatorname{\mathrm{Th}}_S(E) \in \operatorname{\mathrm{SH}}(S). \end{align} $$

Lemma 4.2.2. Let $E \to S$ be a vector bundle of constant odd rank. Then equation (4.2.1.b) is an isomorphism $\operatorname {\mathrm {Th}}_{\mathbb {P}(E)}(\mathcal {O}(-1)) \xrightarrow {\sim } \operatorname {\mathrm {Th}}_S(E)$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

Proof. It follows from (4.1.6) that both vertical arrows in the diagram (4.2.1.a) are isomorphisms in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , hence so the induced map on homotopy cofibers.

Proposition 4.2.3. Let $E \to S$ be a vector bundle of constant odd rank. Then the projection $E^\circ \to S$ admits a section in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , inducing a decomposition in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$

$$\begin{align*}\Sigma^{\infty}_+ E^{\circ} \simeq \mathbf{1}_S \oplus \Sigma^{-2,-1}\operatorname{\mathrm{Th}}_S(E). \end{align*}$$

Proof. By (2.2.6), we have a splitting in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$

$$\begin{align*}\Sigma^{\infty}_+ \mathcal{O}(-1)^\circ \simeq \Sigma^{\infty}_+ \mathbb{P}(E) \oplus \Sigma^{-2,-1}\operatorname{\mathrm{Th}}_{\mathbb{P}(E)}(\mathcal{O}(-1)) \end{align*}$$

and the statement follows from (4.2.1), (4.1.6) and (4.2.2).

Remark 4.2.4. One may deduce that (2.2.7) and (2.2.8) remain valid when L is an odd rank vector bundle instead of a line bundle, using exactly the same arguments but substituting (4.2.3) for (2.2.6).

We deduce the following splitting principle.

Corollary 4.2.5. Let $X \in \operatorname {\mathrm {Sm}}_S$ , and $E \to X$ be a vector bundle of constant odd rank. Then there exists a morphism $f\colon Y \to X$ in $\operatorname {\mathrm {Sm}}_S$ whose image in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ admits a section, and a vector bundle $F \to Y$ such that $f^*E \simeq F \oplus 1$ .

Proof. Applying the functor $\operatorname {\mathrm {SH}}(X)[\eta ^{-1}] \to \operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ of (1.9) we may assume that $X=S$ . Let us denote by $p\colon (E^\vee )^\circ \to S$ the projection. Then $p^*E^\vee $ admits a nowhere vanishing section s. Its dual $s^\vee \colon p^*E \to 1$ is surjective. Letting $Q = \ker s^\vee $ , we have an exact sequence of vector bundles over $(E^\vee )^\circ $

(4.2.5.a) $$ \begin{align} 0 \to Q \to p^*E \to 1 \to 0. \end{align} $$

Then we may find an affine bundle $g \colon Y \to (E^\vee )^\circ $ along which the pullback of the sequence (4.2.5.a) splits (we may take for Y the scheme parametrising the sections of $p^*E^\vee \to Q^\vee $ ; see, e.g., [Rio10, p.243]). Then $\Sigma ^{\infty }_+ g \colon \Sigma ^{\infty }_+ Y \to \Sigma ^{\infty }_+ (E^\vee )^\circ $ is an isomorphism in $\operatorname {\mathrm {SH}}(S)$ , and $\Sigma ^{\infty }_+ p\colon \Sigma ^{\infty }_+ (E^\vee )^\circ \to \mathbf {1}_S$ admits a section in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ by (4.2.3). So we may set $f =p \circ g$ .

4.3 Thom spaces of tensor products by line bundles

We are now in position to slightly improve the result obtained in §3.3.

Lemma 4.3.1. Let $E \to S$ be a vector bundle and $u \in H^0(S,\mathbb {G}_m)$ . Assume that E has constant rank r. Then in the notation of (1.5) and (1.6), we have in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$

$$\begin{align*}\operatorname{\mathrm{Th}}(u \operatorname{\mathrm{id}}_E) =\langle u^r \rangle \colon \operatorname{\mathrm{Th}}_S(E) \to \operatorname{\mathrm{Th}}_S(E). \end{align*}$$

Proof. Since, under the identification $\operatorname {\mathrm {Th}}_S(E\oplus 1) = \operatorname {\mathrm {Th}}_S(E) \wedge \operatorname {\mathrm {Th}}_S(1)$ (see equation (1.3.b)) we have

$$\begin{align*}\operatorname{\mathrm{Th}}(u \operatorname{\mathrm{id}}_{E\oplus 1})= \operatorname{\mathrm{Th}}(u \operatorname{\mathrm{id}}_E) \wedge \operatorname{\mathrm{Th}}(u\operatorname{\mathrm{id}}_1) =\operatorname{\mathrm{Th}}(u \operatorname{\mathrm{id}}_E) \wedge \langle u \rangle, \end{align*}$$

we may replace E with $E\oplus 1$ if necessary and thus assume that r is odd. The $\mathbb {G}_m$ -equivariant isomorphism $E^\circ \simeq \mathcal {O}(-1)^\circ $ (see (4.2.1)) yields a commutative square in $\operatorname {\mathrm {SH}}(S)$

where the vertical arrows coincide and are isomorphisms in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ by (4.2.2). In view of equation (1.6.b), we may replace $E \to S$ with $\mathcal {O}(-1) \to \mathbb {P}(E)$ , and thus assume that E is a line bundle. By (2.2.8), we may replace S with $E^\circ $ , and thus assume that the line bundle $E \to S$ admits a trivialisation $\alpha \colon 1 \xrightarrow {\sim } E$ . Then we have a commutative square of isomorphisms in $\operatorname {\mathrm {SH}}(S)$

By definition $\operatorname {\mathrm {Th}}(u \operatorname {\mathrm {id}}_1) = \langle u \rangle $ , and we deduce using equation (1.6.b) that $\operatorname {\mathrm {Th}}(u \operatorname {\mathrm {id}}_E) = \langle u \rangle \in \operatorname {\mathrm {Aut}}_{\operatorname {\mathrm {SH}}(S)}(\operatorname {\mathrm {Th}}_S(E))$ . Since $\langle u^2 \rangle =\operatorname {\mathrm {id}}$ by (2.1.1) and r is odd, it follows that $\langle u \rangle = \langle u^r \rangle $ , concluding the proof.

Proposition 4.3.2. Let $L \to S$ be a line bundle, and $V \to S$ a vector bundle of constant rank r. If $s \in \mathbb {Z}$ is such that $rs$ is even, then there exists an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$

$$\begin{align*}\operatorname{\mathrm{Th}}_S(V) \simeq \operatorname{\mathrm{Th}}_S(V \otimes L^{\otimes s}). \end{align*}$$

Proof. Upon replacing V with $V \otimes L^{\otimes s}$ , we may assume that $s \geq 0$ . When $\alpha \colon 1 \xrightarrow {\sim }L$ is a trivialisation of the line bundle L over S we have an isomorphism in $\operatorname {\mathrm {SH}}(S)$

(4.3.2.a) $$ \begin{align} \operatorname{\mathrm{Th}}(\operatorname{\mathrm{id}}_V \otimes \alpha^{\otimes s}) \colon \operatorname{\mathrm{Th}}_S(V) \to \operatorname{\mathrm{Th}}_S(V \otimes L^{\otimes s}). \end{align} $$

Any trivialisation of L is of the form $u \alpha $ for some $u \in H^0(S,\mathbb {G}_m)$ , and we have in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , by (4.3.1) and (2.1.1)

$$\begin{align*}\operatorname{\mathrm{Th}}(\operatorname{\mathrm{id}}_V \otimes (u \alpha)^{\otimes s}) = \operatorname{\mathrm{Th}}(\operatorname{\mathrm{id}}_V \otimes \alpha^{\otimes s}) \circ \operatorname{\mathrm{Th}}(u^s\operatorname{\mathrm{id}}_V) = \operatorname{\mathrm{Th}}(\operatorname{\mathrm{id}}_V \otimes \alpha^{\otimes s}) \circ \langle u^{rs} \rangle = \operatorname{\mathrm{Th}}(\operatorname{\mathrm{id}}_V \otimes \alpha^{\otimes s}). \end{align*}$$

It follows that the image of the isomorphism (4.3.2.a) in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ is independent of the choice of the trivialisation $\alpha $ , and we conclude as in the proof of (3.3.1).

5 Classifying spaces and characters

5.1 Models for étale classifying spaces

Here, we recall some facts concerning the geometric models of the étale classifying space of a linear algebraic group given in [MV99, §4.2].

5.1.1

Let G be a linear algebraic group over S. Let $(V_m,U_m,f_m)$ , for $m\in \mathbb {N} \smallsetminus \{0\}$ , be an admissible gadget with a nice (right) G-action, in the sense of [MV99, Definition 4.2.1]. Here, $V_m \to S$ are G-equivariant vector bundles, and $U_m \subset V_m$ are G-invariant open subschemes where the G-action is free. Set $E_mG=U_m$ and $B_mG = (E_mG)/G$ . Let us define $\mathrm {B} G \in \operatorname {\mathrm {Spc}}(S)$ as the colimit of the motivic spaces $B_m G$ as m runs over $\mathbb {N} \smallsetminus \{0\}$ . It is proved in [MV99, Proposition 4.2.6] that the weak-equivalence class of $\mathrm {B} G$ does not depend on the choice of $(V_m,U_m,f_m)$ . More precisely if $(V_m,U_m,f_m),(V^{\prime }_m,U^{\prime }_m,f^{\prime }_m)$ are admissible gadgets with a nice G-action, and $U_m \to U^{\prime }_m$ are G-equivariant morphisms commuting with the morphisms $f_m,f_m'$ , then the induced morphism of motivic spaces $\operatorname {\mathrm {colim}}_m (U_m/G) \to \operatorname {\mathrm {colim}}_m (U^{\prime }_m/G)$ is a weak equivalence. In the sequel, we will refer to a system $(V_m,U_m,f_m)$ as above as a model for $\mathrm {B} G$ and use the notation $E_mG,B_mG$ .

5.1.2

In the situation of (5.1.1), since $B_1G$ is cofibrant and each $B_m G \to B_{m+1} G$ is a cofibration (for the model structure of [MV99]), it follows that the colimit $\mathrm {B} G$ is canonically weakly equivalent to the homotopy colimit of the motivic spaces $B_mG$ in $\operatorname {\mathrm {Spc}}(S)$ (see, e.g., [Hir03, Theorem 19.9.1]).

5.1.3

Let G be a linear algebraic group over S, and choose a model for $\mathrm {B} G$ . Since the map $\operatorname {\mathrm {colim}}_m E_mG \to S$ is a weak equivalence of motivic spaces [MV99, Proposition 4.2.3], we obtain a canonical morphism $S \to \mathrm {B} G$ in $\operatorname {\mathrm {H}}(S)$ . We say that the model is pointed if we are given an S-point of $E_1 G$ . This yields map $S \to \mathrm {B} G$ in $\operatorname {\mathrm {Spc}}(S)$ , whose image in $\operatorname {\mathrm {H}}(S)$ is the canonical morphism described just above.

5.1.4

(See also [MV99, p.133].) Let us fix an integer $n\in \mathbb {N}$ and describe an explicit model for $\operatorname {\mathrm {BGL}}_n$ . Fix an integer $p \geq n$ (we will typically take $p=n$ ). For $s \in \mathbb {N}$ , we denote by $\operatorname {\mathrm {Gr}}(n,s)$ the Grassmannian of rank n subbundles $U\subset 1^{\oplus s}$ over S (for us a subbundle is locally split, so $1^{\oplus s}/U$ is a vector bundle). For each $m\in \mathbb {N} \smallsetminus \{0\}$ , consider the S-scheme $V_{m,p}$ parametrising the vector bundles maps $1^{\oplus n} \to 1^{\oplus pm}$ ; then $V_{m,p} \to S$ is a vector bundle. Let $U_{m,p}$ the open subscheme of $V_{m,p}$ parametrising those vector bundle maps admitting Zariski locally a retraction (i.e., making $1^{\oplus n}$ a subbundle of $1^{\oplus pm}$ ). Then the natural left $\operatorname {\mathrm {GL}}_n$ -action on $1^{\oplus n}$ induces a right $\operatorname {\mathrm {GL}}_n$ -action on $U_{m,p}$ , which is free, and the quotient $U_{m,p}/\operatorname {\mathrm {GL}}_n$ can be identified with the Grassmannian $\operatorname {\mathrm {Gr}}(n,pm)$ . The inclusion $1^{\oplus m} \subset 1^{\oplus m+1}$ given by the vanishing of the last coordinate induces an inclusion

$$\begin{align*}1^{\oplus pm} = (1^{\oplus m})^{\oplus p} \subset (1^{\oplus m+1})^{\oplus p} = 1^{\oplus p(m+1)}, \end{align*}$$

which yields a $\operatorname {\mathrm {GL}}_n$ -equivariant morphism $f_{m,p} \colon U_{m,p} \to U_{m+1,p}$ .

Then the family $(V_{m,p},U_{m,p},f_{m,p})$ is an admissible gadget with a nice $\operatorname {\mathrm {GL}}_n$ -action. Indeed, the first condition of [MV99, Definition 4.2.1] is satisfied because $U_{1,p}$ possesses an S-point, and the second condition is satisfied with $j=2i$ . The fact that the group $\operatorname {\mathrm {GL}}_n$ is special implies the validity of condition (3) of [MV99, Definition 4.2.4]. We thus obtained a model for $\operatorname {\mathrm {BGL}}_n$ . We have just seen that this model is pointed (in the sense of (5.1.3)); a canonical pointing when $p=n$ is induced by the identity of $1^{\oplus n}$ .

5.1.5

Let $H \subset G$ be an inclusion of linear algebraic groups over S. Then any admissible gadget with a nice G-action is also one with a nice H-action (where the H-action is given by restricting the G-action). Indeed, the only nonimmediate point is condition (3) of [MV99, Definition 4.2.4]. So let F be a smooth S-scheme with a free right H-action. Consider the quotient $E=(F \times G)/H$ , where the right H-action on G is given by letting $h \in H$ act via $g \mapsto h^{-1}g$ . Right multiplication in G induces a free right G-action on E. For any $U \in \operatorname {\mathrm {Sm}}_S$ with a right G-action, we have isomorphisms

$$\begin{align*}(E \times U)/G \simeq ((F \times G)/H \times U)/G \simeq (F \times (G\times U)/G)/H \simeq (F\times U)/H, \end{align*}$$

which are functorial in U and thus permit to identify the morphisms $(E \times U)/G \to E/G$ and $(F \times U)/H \to F/H$ . Since the former is an epimorphism in the Nisnevich topology (as the group G is nice), so is the latter.

Thus, given a model for $\mathrm {B} G$ , we obtain a model for $\mathrm {B} H$ , where $E_mH=E_mG$ with the induced H-action. This yields morphisms

$$\begin{align*}B_m H = (E_mH)/H = (E_mG)/H \to (E_mG)/G = B_mG \end{align*}$$

which are compatible with the transition maps as m varies and thus a map $\mathrm {B} H \to \mathrm {B} G$ .

5.1.6

(See also [MV99, Remark 4.2.7].) Assume that G is a linear algebraic group over S, and fix an embedding $G \subset \operatorname {\mathrm {GL}}_n$ as a closed subgroup. By (5.1.5), every (pointed) model for $\operatorname {\mathrm {BGL}}_n$ induces a (pointed) model for $\mathrm {B} G$ . Since $\operatorname {\mathrm {BGL}}_n$ admits a pointed model by (5.1.4), so does $\mathrm {B} G$ .

5.2 Products

5.2.1

Let $G,G'$ be linear algebraic groups over S. Choose admissible gadgets $(V_m,U_m,f_m)$ with a nice G-action, and $(V^{\prime }_m,U^{\prime }_m,f^{\prime }_m)$ with a nice $G'$ -action (in the sense of [MV99, Definition 4.2.1], recall from (5.1.6) that such exist). Then the family $(V_m \times V_m',U_m \times U_m',f_m \times f_m')$ constitutes an admissible gadget with a nice $G \times G'$ -action. Indeed, to check the last condition of [MV99, Definition 4.2.4], let $T \to X$ be a $G\times G'$ -torsor in $\operatorname {\mathrm {Sm}}_S$ . Then the projection $(T \times U \times U')/(G\times G') \to T/(G\times G')$ factors as

$$\begin{align*}(T_1 \times U')/G' \to T_1/G' \end{align*}$$

where $T_1 = (T \times U)/G$ followed by

$$\begin{align*}(T_2 \times U)/G \to T_2/G, \end{align*}$$

where $T_2 = T/G'$ . Each morphism is an epimorphism in the Nisnevich topology by assumption, hence so is their composite.

Under this choice of a model for $\mathrm {B} G$ , we have

(5.2.1.a) $$ \begin{align} B_m(G \times G') = B_mG \times B_mG'. \end{align} $$

Lemma 5.2.2. If $G,G'$ are linear algebraic groups over S, we have an isomorphism

$$\begin{align*}\mathrm{B}(G \times G') \simeq \mathrm{B} G \times \mathrm{B} G' \in \operatorname{\mathrm{H}}(S). \end{align*}$$

Proof. Since the product with a given motivic space commutes with homotopy colimits, we have isomorphisms in $\operatorname {\mathrm {H}}(S)$

$$ \begin{align*} \mathrm{B} (G \times G')&\simeq \operatorname{\mathrm{hocolim}}_m (B_mG \times B_mG') && \text{by equation (5.2.1.a) and (5.1.2)}\\ &\simeq \operatorname{\mathrm{hocolim}}_m \operatorname{\mathrm{hocolim}}_d (B_mG \times B_dG') && \text{by a cofinality argument}\\ &\simeq \operatorname{\mathrm{hocolim}}_m (B_mG \times \operatorname{\mathrm{hocolim}}_d B_dG') \\ &\simeq (\operatorname{\mathrm{hocolim}}_m B_mG) \times (\operatorname{\mathrm{hocolim}}_d B_dG')\\ &\simeq \mathrm{B} G \times \mathrm{B} G'&&\text{by (5.1.2).}\\[-35pt]\end{align*} $$

5.3 Characters

In this section, we discuss general facts relating the classifying space of a linear algebraic group G to that of the kernel H of a character of G, which will be applied to explicit situations in §6.

5.3.1

Let G be a linear algebraic group over S, and fix a model for $\mathrm {B} G$ (see (5.1.1)). Assume given a character of G, that is a morphism of algebraic groups $\chi \colon G \to \mathbb {G}_m$ . Considering the right G-action on ${\mathbb {A}}^1$ given by letting $g\in G$ act via $\lambda \mapsto \chi (g)^{-1} \lambda $ , we define for each $m \in \mathbb {N} \smallsetminus \{0\}$ a line bundle over $B_mG$ :

(5.3.1.a) $$ \begin{align} C_m(\chi)= (E_mG \times {\mathbb{A}}^1)/G. \end{align} $$

The assignment $\chi \mapsto C_m(\chi )$ satisfies

(5.3.1.b) $$ \begin{align} C_m(\chi \chi') = C_m(\chi) \otimes C_m(\chi'), \end{align} $$

yielding group morphisms

$$\begin{align*}\operatorname{\mathrm{Hom}}_{\text{alg. groups}}(G,\mathbb{G}_m) \to \operatorname{\mathrm{Pic}}(B_mG) \quad; \quad \chi \mapsto C_m(\chi). \end{align*}$$

Let now $H \subset G$ be a closed subgroup, and consider the morphisms $B_m H \to B_mG$ defined in (5.1.5). If $\chi |_H$ denotes the restriction of the character $\chi $ to H, then

(5.3.1.c) $$ \begin{align} C_m(\chi) \times_{B_mG} B_mH \simeq C_m(\chi|_H). \end{align} $$

If $G'$ is a linear algebraic group over S, letting $\tilde {\chi }\colon G \times G' \to G \xrightarrow {\chi } \mathbb {G}_m$ be the induced character of $G \times G'$ , we have

(5.3.1.d) $$ \begin{align} C_m(\tilde{\chi}) = C_m(\chi) \times B_mG'. \end{align} $$

5.3.2

Let G be a linear algebraic group over S and $\chi $ a surjective character of G. Letting $H=\ker \chi $ , we thus have an exact sequence of algebraic groups over S

(5.3.2.a) $$ \begin{align} 1 \to H \to G \xrightarrow{\chi} \mathbb{G}_m \to 1. \end{align} $$

Let us fix a model for $\mathrm {B} G$ . As explained in (5.1.5), this yields a model for $\mathrm {B} H$ , and morphisms $p_m \colon B_mH \to B_mG$ for $m \in \mathbb {N} \smallsetminus \{0\}$ . By (5.3.1), we also have a line bundle $C_m(\chi )$ over $B_mG$ such that

$$\begin{align*}C_m(\chi)^{\circ} = ((E_mG \times {\mathbb{A}}^1)/G)^\circ = (E_mG \times \mathbb{G}_m)/G = ((E_mG)/H \times \mathbb{G}_m)/\mathbb{G}_m= (E_mG)/H =B_mH. \end{align*}$$

In view of (1.3.a), this yields a cofiber sequence in ${\mathrm{Spc}_{\bullet }}(S)$ , for each $m\in \mathbb {N}\smallsetminus \{0\}$

(5.3.2.b) $$ \begin{align} (B_mH)_+ \xrightarrow{{p_m}_+} (B_mG)_+ \to \operatorname{\mathrm{Th}}_{B_mG}(C_m(\chi)). \end{align} $$

More generally (as in (1.4)), if $V \to B_mG$ is a vector bundle, we have a cofiber sequence in ${\mathrm{Spc}_{\bullet }}(S)$ ,

(5.3.2.c) $$ \begin{align} \operatorname{\mathrm{Th}}_{B_mH}(V) \to \operatorname{\mathrm{Th}}_{B_mG}(V) \to \operatorname{\mathrm{Th}}_{B_mG}(C_m(\chi) \oplus V). \end{align} $$

5.3.3

In the situation of (5.3.2), let us define

$$\begin{align*}\operatorname{\mathrm{Th}}_{\mathrm{B} G}(C(\chi)) =\operatorname{\mathrm{colim}}_m \operatorname{\mathrm{Th}}_{B_mG}(C_m(\chi)) \in {\mathrm{Spc}_{\bullet}}(S). \end{align*}$$

As in (5.1.2), this coincides with the homotopy colimit (the transition morphisms are again monomorphisms, being directed colimits of such). We will also write $\operatorname {\mathrm {Th}}_{\mathrm {B} G}(C(\chi )) \in \operatorname {\mathrm {Spt}}(S)$ instead of $\Sigma ^{\infty } \operatorname {\mathrm {Th}}_{\mathrm {B} G}(C(\chi ))$ . Taking the (homotopy) colimit of equation (5.3.2.b) yields a cofiber sequence in ${\mathrm{Spc}_{\bullet }}(S)$

(5.3.3.a) $$ \begin{align} (\mathrm{B} H)_+ \to (\mathrm{B} G)_+ \to \operatorname{\mathrm{Th}}_{\mathrm{B} G}(C(\chi)). \end{align} $$

5.3.4

In the situation of (5.3.2), assume that the model for $\mathrm {B} G$ is pointed. Then we have a commutative diagram of S-schemes with Cartesian squares

(5.3.4.a)

where $e_1$ is induced by the S-point and the G-action on $E_1G$ , and $j_1$ , resp. $i_1$ , is obtained by taking the H-quotient, resp. G-quotient of $e_1$ . Composing $i_1$ and $j_1$ with the natural maps $B_1 G \to \mathrm {B} G$ and $B_1 H \to \mathrm {B} H$ respectively, we obtain maps $i \colon S \to \mathrm {B} G$ and $j \colon \mathbb {G}_m \to \mathrm {B} H$ in $\operatorname {\mathrm {Spc}}(S)$ . Note that i is the map described in (5.1.3) and that the left-hand Cartesian square in equation (5.3.4.a) shows that the map $j \colon \mathbb {G}_m \to \mathrm {B} H$ classifies the H-torsor $\chi \colon G \to \mathbb {G}_m$ .

The right-hand Cartesian square in equation (5.3.4.a) shows that the $\mathbb {G}_m$ -torsor $B_1 H \to B_1G$ pulls back to the trivial torsor along $i_1$ , which yields a trivialisation of the line bundle $i_1^*C_1(\chi )$ over S and thus a morphism in ${\mathrm{Spc}_{\bullet }}(S)$

$$\begin{align*}t \colon T=\operatorname{\mathrm{Th}}_S(1) \to \operatorname{\mathrm{Th}}_{B_1G}(C_1(\chi)) \to \operatorname{\mathrm{Th}}_{\mathrm{B} G}(C(\chi)). \end{align*}$$

We thus obtain a commutative diagram in ${\mathrm{Spc}_{\bullet }}(S)$ , whose rows are cofiber sequences

(5.3.4.b)

5.3.5

In the situation of (5.3.2), using (2.2.5) for the line bundle $C_m(\chi ) \to B_mG$ , and applying the functor ${\mathrm{Spc}_{\bullet }}(B_mG)[\eta ^{-1}] \to {\mathrm{Spc}_{\bullet }}(S)[\eta ^{-1}]$ of (1.9), we have homotopy cocartesian squares in ${\mathrm{Spc}_{\bullet }}(S)[\eta ^{-1}]$

which are compatible with the transition maps as m varies by (2.2.3). Taking the homotopy colimit, and proceeding as in the proof of (2.2.6), we obtain an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$

(5.3.5.a) $$ \begin{align} \Sigma^{\infty}_+ \mathrm{B} H \simeq \Sigma^{\infty}_+ \mathrm{B} G \oplus \Sigma^{-2,-1}\operatorname{\mathrm{Th}}_{\mathrm{B} G}(C(\chi)) \in \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align} $$

6 Computations of classifying spaces

6.1 Diagonalisable groups

Using the embeddings $\mathbb {P}^k \subset \mathbb {P}^{k+1}$ of (4.1.1) for $k \in \mathbb {N}$ , we define, for $n \in \mathbb {Z}$

$$\begin{align*}\mathbb{P}^\infty = \operatorname{\mathrm{hocolim}}_k \mathbb{P}^k \in \operatorname{\mathrm{Spc}}(S) \quad \text{and} \quad \operatorname{\mathrm{Th}}_{\mathbb{P}^{\infty}}(\mathcal{O}(n)) = \operatorname{\mathrm{hocolim}}_k \operatorname{\mathrm{Th}}_{\mathbb{P}^k}(\mathcal{O}(n)) \in {\mathrm{Spc}_{\bullet}}(S), \end{align*}$$

and as usual write $\operatorname {\mathrm {Th}}_{\mathbb {P}^\infty }(\mathcal {O}(n)) \in \operatorname {\mathrm {Spt}}(S)$ instead of $\Sigma ^{\infty } \operatorname {\mathrm {Th}}_{\mathbb {P}^\infty }(\mathcal {O}(n))$ . We have a natural map $\iota _\infty \colon S=\mathbb {P}^0 \to \mathbb {P}^\infty $ in $\operatorname {\mathrm {Spc}}(S)$ . For each $n \in \mathbb {Z}$ , the line bundle $\mathcal {O}(n)$ over $\mathbb {P}^0$ admits a canonical trivialisation so that equation (4.1.2.a) yields a canonical map in $\operatorname {\mathrm {Spt}}(S)$

(6.1.0.a) $$ \begin{align} \Sigma^{2,1} \mathbf{1}_S \to \operatorname{\mathrm{Th}}_{\mathbb{P}^\infty}(\mathcal{O}(n)). \end{align} $$

Proposition 6.1.1. Let $n\in \mathbb {Z}$ . The following hold in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ :

1. (i) The morphism $\iota _\infty $ induces an isomorphism $\mathbf {1}_S \simeq \Sigma ^{\infty }_+\mathbb {P}^\infty $ .

2. (ii) If n is odd, then $\operatorname {\mathrm {Th}}_{\mathbb {P}^\infty }(\mathcal {O}(n)) = 0$ .

3. (iii) If n is even, then equation (6.1.0.a) induces an isomorphism $\Sigma ^{2,1}\mathbf {1}_S \simeq \operatorname {\mathrm {Th}}_{\mathbb {P}^\infty }(\mathcal {O}(n))$ .

Proof. We apply (4.1.3) with $\mathcal {D} = (n)$ , and so $\mathcal {O}(\mathcal {D})=\mathcal {O}(n)$ . We obtain that $\Sigma ^{2,1}\mathbf {1}_S \to \operatorname {\mathrm {Th}}_{\mathbb {P}^k}(\mathcal {O}(n))$ is a weak equivalence in $\operatorname {\mathrm {Spt}}(S)[\eta ^{-1}]$ when $n,k$ are even. Taking the homotopy colimit over k yields a weak equivalence $\Sigma ^{2,1}\mathbf {1}_S \to \operatorname {\mathrm {Th}}_{\mathbb {P}^\infty }(\mathcal {O}(n))$ in $\operatorname {\mathrm {Spt}}(S)[\eta ^{-1}]$ when n is even. This proves equation (iii). The other statements are deduced in a similar way from (4.1.3).

6.1.2

Consider the model for $\mathrm {B} \mathbb {G}_m$ described in (5.1.4) with $p=n=1$ , under the identification $\mathbb {G}_m = \operatorname {\mathrm {GL}}_1$ . Then $B_m\mathbb {G}_m = \mathbb {P}^{m-1}$ , and thus $\mathrm {B} \mathbb {G}_m = \mathbb {P}^{\infty }$ . Furthermore, the line bundle $C_m(\operatorname {\mathrm {id}}_{\mathbb {G}_m})$ over $B_mG$ defined in equation (5.3.1.a) may be identified with the tautological bundle $\mathcal {O}(-1)$ over $\mathbb {P}^{m-1}$ .

Theorem 6.1.3. Let $n\in \mathbb {N} \smallsetminus \{0\}$ . The following hold in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ :

1. (i) The natural morphism $\mathbf {1}_S \to \Sigma ^{\infty }_+\mathrm {B}\mathbb {G}_m$ is an isomorphism.

2. (ii) If n is odd, the natural morphism $\mathbf {1}_S \to \Sigma ^{\infty }_+\mathrm {B}\mu _n$ is an isomorphism.

3. (iii) If n is even, the morphism $\mathbb {G}_m \to \mathrm {B} \mu _n$ classifying the $\mu _n$ -torsor $\mathbb {G}_m \to \mathbb {G}_m$ given by taking n-th powers induces an isomorphism $\Sigma ^{\infty }_+\mathbb {G}_m \simeq \Sigma ^{\infty }_+\mathrm {B}\mu _n$ .

Proof. Let us consider the model for $\mathrm {B}\mathbb {G}_m$ described in (6.1.2), where $B_m\mathbb {G}_m = \mathbb {P}^{m-1}$ and $\mathrm {B} \mathbb {G}_m = \mathbb {P}^\infty $ . Then the first statement follows from (6.1.1.i).

Next, consider the character $n \colon \mathbb {G}_m \to \mathbb {G}_m$ given by taking n-th powers. Its kernel is $\mu _n$ , and the line bundle $C_m(n)$ over $B_m\mathbb {G}_m$ (defined in equation (5.3.1.a)) corresponds to the line bundle $\mathcal {O}(-n)$ over $\mathbb {P}^{m-1}$ (this may be seen for instance by combining (6.1.2) with equation (5.3.1.b)). So we are in the situation of (5.3.2) with $G=\mathbb {G}_m,\chi =n,H=\mu _n$ . Thus, equation (5.3.3.a) yields a distinguished triangle in $\operatorname {\mathrm {SH}}(S)$

$$\begin{align*}\Sigma^{\infty}_+ \mathrm{B} \mu_n \to \Sigma^{\infty}_+ \mathrm{B} \mathbb{G}_m \to \operatorname{\mathrm{Th}}_{\mathbb{P}^\infty}(\mathcal{O}(-n)) \to \Sigma^{1,0} \Sigma^{\infty}_+ \mathrm{B} \mu_n. \end{align*}$$

If n is odd, then $\operatorname {\mathrm {Th}}_{\mathbb {P}^\infty }(\mathcal {O}(-n))=0$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ by (6.1.1.ii), and the above distinguished triangle shows that the morphism $\Sigma ^{\infty }_+ \mathrm {B} \mu _n \to \Sigma ^{\infty }_+ \mathrm {B} \mathbb {G}_m$ is an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ . Thus, the second statement follows from the first.

Assume that n is even. Then in the diagram (5.3.4.b) the maps $i_+$ and t become isomorphisms in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ by (6.1.1.i) and (6.1.1.iii), hence $\Sigma ^{\infty }_+ j \colon \Sigma ^{\infty }_+ \mathbb {G}_m \to \Sigma ^{\infty }_+ \mathrm {B} \mu _n$ is also one. As observed in (5.3.2), the latter is induced by the $\mu _n$ -torsor $n\colon \mathbb {G}_m \to \mathbb {G}_m$ .

Remark 6.1.4. Combining (6.1.3) with (5.2.2), we have thus obtained a ‘computation’ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ of the classifying space $\mathrm {B} G$ of every finitely generated diagonalisable group G.

6.2 $\operatorname {\mathrm {SL}}$ versus $\operatorname {\mathrm {SL}}^c$

Let $n \in \mathbb {N} \smallsetminus \{0\}$ . Consider the character

$$\begin{align*}\nu_n \colon \operatorname{\mathrm{GL}}_n \times \mathbb{G}_m \to \mathbb{G}_m \quad; \quad (M,t) \mapsto t^{-2} \det M. \end{align*}$$

By definition (see [PW18, §3]), we have $\operatorname {\mathrm {SL}}_n^c = \ker \nu _n$ . We view $\operatorname {\mathrm {SL}}_n$ as a subgroup of $\operatorname {\mathrm {SL}}^c_n$ via the mapping $M \mapsto (M,1)$ .

Proposition 6.2.1. For $n \in \mathbb {N}\smallsetminus \{0\}$ , the inclusion $\operatorname {\mathrm {SL}}_n \subset \operatorname {\mathrm {SL}}_n^c$ induces an isomorphism

$$\begin{align*}\Sigma^{\infty}_+ \operatorname{\mathrm{BSL}}_n \xrightarrow{\sim} \Sigma^{\infty}_+ \operatorname{\mathrm{BSL}}^c_n \text{ in } \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align*}$$

Proof. The character

$$\begin{align*}\delta_n \colon \operatorname{\mathrm{SL}}_n^c \to \mathbb{G}_m \quad; \quad (M,t) \mapsto t \end{align*}$$

is surjective (recall that $n \geq 1$ ), and satisfies $\operatorname {\mathrm {SL}}_n = \ker \delta _n$ .

Set $P_n=\operatorname {\mathrm {GL}}_n \times \mathbb {G}_m$ , and denote by $q_n\colon P_n \to \mathbb {G}_m$ the second projection. Let us fix an arbitrary model for $\operatorname {\mathrm {BGL}}_n$ , but choose the model for $\mathrm {B}\mathbb {G}_m$ described in (6.1.2) so that $B_m \mathbb {G}_m=\mathbb {P}^{m-1}$ . Recall from (5.2.1.a) that this yields a model for $\mathrm {B} P_n$ such that

$$\begin{align*}B_mP_n = (B_m\operatorname{\mathrm{GL}}_n) \times (B_m\mathbb{G}_m) = (B_m\operatorname{\mathrm{GL}}_n) \times \mathbb{P}^{m-1}. \end{align*}$$

Letting $g_n \colon P_n \to \operatorname {\mathrm {GL}}_n$ be the first projection, we have, as characters $P_n \to \mathbb {G}_m$

$$\begin{align*}\nu_n = g_n^*({\det}_n) \cdot q_n^*(\operatorname{\mathrm{id}}_{\mathbb{G}_m})^{-2}, \end{align*}$$

where $\det _n \colon \operatorname {\mathrm {GL}}_n \to \mathbb {G}_m$ denotes the determinant morphism. It follows from equations (5.3.1.b) and (5.3.1.d) that, as line bundles over $B_m P_n$ , we have in the notation of (5.3.1.a)

$$\begin{align*}C_m(\nu_n) \simeq C_m({\det}_{n}) \boxtimes C_m(\operatorname{\mathrm{id}}_{\mathbb{G}_m})^{\otimes -2} \quad; \quad C_m(q_n) \simeq 1 \boxtimes C_m(\operatorname{\mathrm{id}}_{\mathbb{G}_m}). \end{align*}$$

Recall from (6.1.2) that the line bundle $C_m(\operatorname {\mathrm {id}}_{\mathbb {G}_m}) \to B_m\mathbb {G}_m$ corresponds to $\mathcal {O}(-1) \to \mathbb {P}^{m-1}$ . Applying (4.1.5) to the projective bundle $B_mP_n \to B_m \operatorname {\mathrm {GL}}_n$ , for m even we have

(6.2.1.a) $$ \begin{align} \operatorname{\mathrm{Th}}_{B_mP_n}(C_m(\nu_n) \oplus C_m(q_n))=0= \operatorname{\mathrm{Th}}_{B_mP_n}(C_m(q_n)) \in \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align} $$

Since the character $\delta _n$ is the restriction of $q_n \colon P_n \to \mathbb {G}_m$ , it follows from equation (5.3.1.c) that the line bundle $C_m(\delta _n)$ over $B_m\operatorname {\mathrm {SL}}_n^c$ is the pullback of $C_m(q_n)$ over $B_mP_n$ . By (5.3.2.c), we have a distinguished triangle in $\operatorname {\mathrm {SH}}(S)$

$$ \begin{align*} \operatorname{\mathrm{Th}}_{B_m\operatorname{\mathrm{SL}}_n^c}(C_m(\delta_n)) \to \operatorname{\mathrm{Th}}_{B_mP_n}(C_m(q_n)) \to& \operatorname{\mathrm{Th}}_{B_mP_n}(C_m(\nu_n) \oplus C_m(q_n)) \\ &\quad \quad \to \Sigma^{1,0}\operatorname{\mathrm{Th}}_{B_m\operatorname{\mathrm{SL}}_n^c}(C_m(\delta_n)) \end{align*} $$

so that, in view of equation (6.2.1.a)

$$\begin{align*}\operatorname{\mathrm{Th}}_{B_m\operatorname{\mathrm{SL}}_n^c}(C_m(\delta_n)) =0 \in \operatorname{\mathrm{SH}}(S)[\eta^{-1}] \quad \text{for } m \text{ even}. \end{align*}$$

Now, the distinguished triangle in $\operatorname {\mathrm {SH}}(S)$ (see (5.3.2.b))

$$\begin{align*}\Sigma^{\infty}_+ B_m\operatorname{\mathrm{SL}}_n \to \Sigma^{\infty}_+ B_m\operatorname{\mathrm{SL}}_n^c \to \operatorname{\mathrm{Th}}_{B_m\operatorname{\mathrm{SL}}_n^c}(C_m(\delta_n)) \to \Sigma^{1,0} \Sigma^{\infty}_+ B_m\operatorname{\mathrm{SL}}_n \end{align*}$$

implies that, for m even, the natural map induces an isomorphism

$$\begin{align*}\Sigma^{\infty}_+ B_m\operatorname{\mathrm{SL}}_n \xrightarrow{\sim} \Sigma^{\infty}_+ B_m\operatorname{\mathrm{SL}}_n^c \quad \text{in } \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align*}$$

The statement follows by taking the homotopy colimit.

Remark 6.2.2. Let $A \in \operatorname {\mathrm {SH}}(S)$ be an $\eta $ -periodic commutative ring spectrum, and consider the corresponding cohomology theory $A^{*,*}(-)$ (see (3.1.1)). Then by (6.2.1), we have a natural isomorphism

$$\begin{align*}A^{*,*}(\operatorname{\mathrm{BSL}}_n^c) \simeq A^{*,*}(\operatorname{\mathrm{BSL}}_n). \end{align*}$$

If A is $\operatorname {\mathrm {SL}}$ -oriented (see (3.2.5)) and $S=\operatorname {\mathrm {Spec}} k$ with k a field of characteristic not two, Ananyevskiy computed in [Ana15, Theorem 10] that

$$\begin{align*}A^{*,*}(\operatorname{\mathrm{BSL}}_n) = \begin{cases} A^{*,*}(S)[[p_1,\ldots,p_{r-1},e]]_h & \text{if } n=2r \text{ with } r\in\mathbb{N} \smallsetminus \{0\}\\ A^{*,*}(S)[[p_1,\ldots,p_r]]_h & \text{if } n=2r+1 \text{ with } r\in \mathbb{N}, \end{cases} \end{align*}$$

where $p_i$ has degree $(4i,2i)$ and e has degree $(2r,r)$ (here, the notation $R[[x_1,\ldots ,x_m]]_h$ refers to the homogeneous power series ring in m variables over the graded ring R; see [Ana15, Definition 27]). This computation remains valid (with exactly the same arguments) when S is an arbitrary noetherian scheme of finite dimension under the assumption that $2$ is invertible in S. Removing that last assumption seems to require a modification of the arguments of [Ana15], the problem being with [Ana15, Lemma 6] (which is used to prove [Ana15, Theorem 9]).

6.3 $\operatorname {\mathrm {GL}}$ and $\operatorname {\mathrm {SL}}$

In this section, we compare the classifying spaces $\operatorname {\mathrm {BGL}}_{2r}$ , $\operatorname {\mathrm {BGL}}_{2r+1}$ , $\operatorname {\mathrm {BSL}}_{2r}$ , $\operatorname {\mathrm {BSL}}_{2r+1}$ .

6.3.1

Recall that under the model for $\operatorname {\mathrm {BGL}}_n$ described in (5.1.4) for $p=n$ , the scheme $B_m \operatorname {\mathrm {GL}}_n$ is identified with the Grassmannian $\operatorname {\mathrm {Gr}}(n,nm)$ . The closed immersion $\operatorname {\mathrm {Gr}}(n,nm) \to \operatorname {\mathrm {Gr}}(n+1,(n+1)m)$ mapping a subbundle $E \subset (1^{\oplus m})^{\oplus n}$ to $E \oplus 1 \subset (1^{\oplus m})^{\oplus n} \oplus 1^{\oplus m} = (1^{\oplus m})^{\oplus n+1}$ , where the inclusion $1 \subset 1^{\oplus m}$ is given by the vanishing of the last $m-1$ coordinates, induces a morphism $f_m \colon B_m \operatorname {\mathrm {GL}}_n \to B_m\operatorname {\mathrm {GL}}_{n+1}$ which is compatible with the transition maps as m varies. This yields a morphism in $\operatorname {\mathrm {Spc}}(S)$

(6.3.1.a) $$ \begin{align} \operatorname{\mathrm{BGL}}_n \to \operatorname{\mathrm{BGL}}_{n+1}. \end{align} $$

6.3.2

For integers $u,v,w \in \mathbb {N}$ , we denote by $\operatorname {\mathrm {Gr}}(u\subset v,w)$ the flag variety of subbundles $P \subset Q \subset 1^{\oplus w}$ with $\operatorname {\mathrm {rank}} P=u$ and $\operatorname {\mathrm {rank}} Q=v$ . Let $r,s \in \mathbb {N}$ , and consider the morphisms

$$\begin{align*}\operatorname{\mathrm{Gr}}(2r,s) \xleftarrow{p} \operatorname{\mathrm{Gr}}(2r\subset 2r+1,s) \xrightarrow{q} \operatorname{\mathrm{Gr}}(2r+1,s) \end{align*}$$

given by mapping a flag $P \subset Q$ to P, resp. Q.

For $n \in \{2r,2r+1\}$ , let us denote by $\mathcal {U}_n \subset 1^{\oplus s}$ the tautological rank n subbundle over $\operatorname {\mathrm {Gr}}(n,s)$ , and write $\mathcal {Q}_n = 1^{\oplus s}/\mathcal {U}_n$ . Then the morphism p may be identified with the projective bundle $\mathbb {P}(\mathcal {Q}_{2r})$ , and the morphism q is the projective bundle $\mathbb {P}(\mathcal {U}_{2r+1}^\vee )$ .

Proposition 6.3.3. The map $\operatorname {\mathrm {BGL}}_{2r} \to \operatorname {\mathrm {BGL}}_{2r+1}$ of equation (6.3.1.a) becomes an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ .

Proof. Let $n=2r$ . For $m\in \mathbb {N} \smallsetminus \{0\}$ , consider the commutative diagram in $\operatorname {\mathrm {Sm}}_S$

where the morphism $j_m$ is given by mapping $E \subset (1^{\oplus m})^{\oplus n}$ to

$$\begin{align*}E \subset E \oplus 1 \subset (1^{\oplus m})^{\oplus n} \oplus 1^{\oplus m} = (1^{\oplus m})^{\oplus n+1}, \end{align*}$$

with the inclusion $1 \subset 1^{\oplus m}$ given by the vanishing of the $m-1$ last coordinates. Here, the morphisms $p_m,q_m$ are the morphisms $p,q$ described in (6.3.2) when $s=(n+1)m$ . The morphism $f_m$ is the one described in (6.3.1), and the morphism $g_m$ is induced by the inclusion

(6.3.3.a) $$ \begin{align} (1^{\oplus m})^{\oplus n} =(1^{\oplus m})^{\oplus n} \oplus 0 \subset (1^{\oplus m})^{\oplus n} \oplus 1^{\oplus m} = (1^{\oplus m})^{\oplus n+1}. \end{align} $$

The morphisms of this diagram are compatible with the transition maps as m varies, induced by the inclusions $1^{\oplus m} \subset 1^{\oplus m+1}$ given by the vanishing of the last coordinate.

The morphism $q_m$ is a $\mathbb {P}^n$ -bundle, hence is an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ by (4.1.6) (recall that $n=2r$ is even). The morphism $p_m$ is a $\mathbb {P}^{(n+1)m-n-1}$ -bundle, hence is also an isomorphism in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ when m is odd by (4.1.6).

In the notation of (5.1.4), the morphism $g_m$ is the $\operatorname {\mathrm {GL}}_n$ -quotient of the morphism $U_{m,n} \to U_{m,n+1}$ induced by equation (6.3.3.a). Therefore, it follows from (5.1.1) that the map $\operatorname {\mathrm {colim}}_m g_m$ is a weak equivalence of motivic spaces.

Applying the functor $\Sigma ^{\infty }_+ \colon \operatorname {\mathrm {Sm}}_S \to \operatorname {\mathrm {Spt}}(S)[\eta ^{-1}]$ to the above diagram and taking the homotopy colimit over m, we thus obtain a commutative diagram in $\operatorname {\mathrm {Spt}}(S)[\eta ^{-1}]$ , where all maps are weak equivalences. Since the map (6.3.1.a) is obtained as $\operatorname {\mathrm {colim}}_m f_m$ , the proposition follows.

6.3.4

Let $n \in \mathbb {N} \smallsetminus \{0\}$ . The group $\operatorname {\mathrm {SL}}_n$ is the kernel of the determinant morphism $\det _n \colon \operatorname {\mathrm {GL}}_n \to \mathbb {G}_m$ , which is surjective (as $n\geq 1$ ). We are thus in the situation of (5.3.2) so that we have by equation (5.3.5.a) a splitting

(6.3.4.a) $$ \begin{align} \Sigma^{\infty}_+ \operatorname{\mathrm{BSL}}_n = \Sigma^{\infty}_+ \operatorname{\mathrm{BGL}}_n \oplus \Sigma^{-2,-1}\operatorname{\mathrm{Th}}_{\operatorname{\mathrm{BGL}}_n}(C({\det}_n)) \in \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align} $$

6.3.5

Using the model for $\operatorname {\mathrm {BGL}}_n$ described in (5.1.4) with $p=n$ , the variety $B_m \operatorname {\mathrm {GL}}_n$ coincides with the Grassmannian $\operatorname {\mathrm {Gr}}(n,nm)$ . Observe that the tautological bundle $\mathcal {U}_n$ over this variety is isomorphic to the quotient $(E_m\operatorname {\mathrm {GL}}_n \times {\mathbb {A}}^n)/\operatorname {\mathrm {GL}}_n$ , where the right $\operatorname {\mathrm {GL}}_n$ -action on ${\mathbb {A}}^n$ is given by letting $\varphi \in \operatorname {\mathrm {GL}}_n$ act via $v \mapsto \varphi ^{-1}(v)$ . The $\operatorname {\mathrm {GL}}_n$ -equivariant isomorphism $\det (E_m\operatorname {\mathrm {GL}}_n \times {\mathbb {A}}^n) \simeq E_m\operatorname {\mathrm {GL}}_n \times {\mathbb {A}}^1$ , where the right $\operatorname {\mathrm {GL}}_n$ -action on ${\mathbb {A}}^1$ is given by letting $\varphi \in \operatorname {\mathrm {GL}}_n$ act via $\lambda \mapsto \det _n(\varphi ^{-1}) \lambda $ , yields an isomorphism of line bundles over $B_m\operatorname {\mathrm {GL}}_n$

(6.3.5.a) $$ \begin{align} \det \mathcal{U}_n \simeq C_m({\det}_n). \end{align} $$

Proposition 6.3.6. Let $r \in \mathbb {N}$ . Then in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , the natural morphism $\operatorname {\mathrm {BSL}}_{2r} \to \operatorname {\mathrm {BGL}}_{2r}$ acquires a section, and the natural morphism $\operatorname {\mathrm {BSL}}_{2r+1} \to \operatorname {\mathrm {BGL}}_{2r+1}$ becomes an isomorphism.

Proof. The first statement follows from (6.3.4.a). Let us prove the second. We use the model for $\operatorname {\mathrm {BGL}}_{2r+1}$ described in (5.1.4) with $n=p=2r+1$ so that $B_m \operatorname {\mathrm {GL}}_{2r+1} = \operatorname {\mathrm {Gr}}(2r+1,s)$ where $s=(2r+1)m$ . We consider the situation of (6.3.2) and use the notation thereof. We have an exact sequence of vector bundles over $Y_m=\operatorname {\mathrm {Gr}}(2r \subset 2r+1,s)=\mathbb {P}(\mathcal {Q}_{2r})$

$$\begin{align*}0 \to p^*\mathcal{U}_{2r} \to q^*\mathcal{U}_{2r+1} \to \mathcal{O}_{\mathbb{P}(\mathcal{Q}_{2r})}(-1) \to 0. \end{align*}$$

Taking determinants and using equation (6.3.5.a), we obtain an isomorphism of line bundles

(6.3.6.a) $$ \begin{align} p^*(\det \mathcal{U}_{2r}) \otimes \mathcal{O}_{\mathbb{P}(\mathcal{Q}_{2r})}(-1) \simeq q^*(\det \mathcal{U}_{2r+1}) \simeq q^*C_m({\det}_{2r+1}). \end{align} $$

When m is even, the vector bundle $\mathcal {Q}_{2r}=1^{\oplus s}/\mathcal {U}_{2r}$ has even rank, hence it follows from (4.1.5) and equation (6.3.6.a) that $\operatorname {\mathrm {Th}}_{Y_m}(q^*C_m(\det _{2r+1}))=0$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ . Since $q \colon Y_m \to \operatorname {\mathrm {Gr}}(2r+1,s)$ is a $\mathbb {P}^{2r}$ -bundle, it then follows from (4.1.6) (applied with $V=C_m(\det _{2r+1})$ ) that $\operatorname {\mathrm {Th}}_{\operatorname {\mathrm {Gr}}(2r+1,s)}(C_m(\det _{2r+1}))=0$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ for m even. Taking the homotopy colimit over m, we deduce that $\operatorname {\mathrm {Th}}_{\operatorname {\mathrm {BGL}}_{2r+1}}(C(\det _{2r+1}))=0$ in $\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$ , and the second statement of the proposition follows from equation (6.3.4.a).

Remark 6.3.7. Let $A \in \operatorname {\mathrm {SH}}(S)$ be an $\eta $ -periodic $\operatorname {\mathrm {SL}}$ -oriented commutative ring spectrum (see (3.2.5)), and consider the corresponding cohomology theory $A^{*,*}(-)$ (see (3.1.1)). Assume that $2$ is invertible in S. Combining (6.3.6) and (6.3.3) with Ananyevskiy’s computation of $A^{*,*}(\operatorname {\mathrm {BSL}}_{2r+1})$ (see (6.2.2)), we recover Levine’s computation [Lev19, Theorem 4.1]

$$\begin{align*}A^{*,*}(\operatorname{\mathrm{BGL}}_{2r}) = A^{*,*}(\operatorname{\mathrm{BGL}}_{2r+1}) = A^{*,*}(S)[[p_1,\ldots,p_r]]_h. \end{align*}$$

(Note that this permits to remove some of the technical assumptions present in the statement of [Lev19, Theorem 4.1].)