Introduction
Around 40 years ago, Arason computed the Witt groups of projective spaces [Reference ArasonAra80]. This computation was later revisited by Gille [Reference GilleGil01], Walter [Reference WalterWal03] and Nenashev [Reference NenashevNen09]. 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 [Reference AnanyevskiyAna16a] 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 [Reference AnanyevskiyAna16a] 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 [Reference Bachmann and HopkinsBH20]. 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 [Reference Panin and WalterPW18, §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 [Reference AnanyevskiyAna20, 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 [Reference AnanyevskiyAna20, 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.
-
(i) The projection
$E^\circ \to X$
admits a section in
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
. -
(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 [Reference HautionHau22] 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 [Reference Morel and VoevodskyMV99]. 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, [Reference Panin, Pimenov and RöndigsPPR09, 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,
$$ \begin{align} (E^\circ)_+ \xrightarrow{p_+} X_+ \to \operatorname{\mathrm{Th}}_X(E). \end{align} $$
If
$F \to S$
is a vector bundle and
$f \colon X \to S$
the structural morphism, we have by [Reference Morel and VoevodskyMV99, Proposition 3.2.17 (1)] a natural identification in
${\mathrm{Spc}_{\bullet }}(S)$
$$ \begin{align} \operatorname{\mathrm{Th}}_X(E \oplus f^*F) = \operatorname{\mathrm{Th}}_X(E) \wedge \operatorname{\mathrm{Th}}_S(F). \end{align} $$
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., [Reference Morel and VoevodskyMV99, Theorem 3.2.23]) yields a cofiber sequence in
${\mathrm{Spc}_{\bullet }}(S)$
$$ \begin{align} U_+ \xrightarrow{u_+} X_+ \to \operatorname{\mathrm{Th}}_Y(N). \end{align} $$
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
$$ \begin{align} \operatorname{\mathrm{Th}}(\psi \circ \varphi) = \operatorname{\mathrm{Th}}(\psi) \circ \operatorname{\mathrm{Th}}(\varphi). \end{align} $$
If
$X=S$
and
$f\in \operatorname {\mathrm {End}}_{\operatorname {\mathrm {SH}}(S)}(\mathbf {1}_S)$
, then we have in
$\operatorname {\mathrm {SH}}(S)$
$$ \begin{align} (\Sigma^Ff) \circ \operatorname{\mathrm{Th}}(\varphi) = \operatorname{\mathrm{Th}}(\varphi) \circ (\Sigma^Ef) \colon \operatorname{\mathrm{Th}}_S(E) \to \operatorname{\mathrm{Th}}_S(F). \end{align} $$
(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 [Reference MorelMor04, 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
$$ \begin{align} \langle uv \rangle = \langle u \rangle \circ \langle v \rangle \quad \text{ for } u,v \in H^0(X,\mathbb{G}_m). \end{align} $$
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
$$ \begin{align} f \circ \langle u \rangle = \langle u \rangle \circ f. \end{align} $$
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 [Reference BachmannBac18, §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
$$ \begin{align} A \wedge \Sigma^{\infty} ({\mathbb{A}}^2 \smallsetminus \{0\}) \xrightarrow{\operatorname{\mathrm{id}} \wedge \Sigma^{\infty} \eta} A \wedge \Sigma^{\infty} \mathbb{P}^1 \end{align} $$
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., [Reference Dugger, Hollander and IsaksenDHI04].) 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
$$ \begin{align} T = {\mathbb{A}}^1/\mathbb{G}_m \xrightarrow{\sim} \mathbb{P}^1/{\mathbb{A}}^1 \xleftarrow{\sim} \mathbb{P}^1, \end{align} $$
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., [Reference AnanyevskiyAna20, 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., [Reference AnanyevskiyAna16a, 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
$$ \begin{align} {\mathbb{A}}^2 \smallsetminus \{0\} \to T \wedge (\mathbb{G}_m)_+. \end{align} $$
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 [Reference AnanyevskiyAna16b, 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
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 [Reference HirschhornHir03, 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 [Reference LevineLev19, 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., [Reference HoveyHov99, 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.
-
(i) The natural map
$\operatorname {\mathrm {Th}}_{L^\circ }(V) \to \operatorname {\mathrm {Th}}_S(V)$
extends to a natural isomorphism in
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
$$\begin{align*}\operatorname{\mathrm{Th}}_{L^{\circ}}(V) \simeq \operatorname{\mathrm{Th}}_S(V) \oplus \Sigma^{-2,-1}\operatorname{\mathrm{Th}}_S(V \oplus L). \end{align*}$$
-
(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.
-
(i) Denoting by
$p \colon L^\circ \to X$
the projection, we have a split short exact sequence
$$\begin{align*}0 \to A^{*,*}(X;V) \xrightarrow{p^*} A^{*,*}(L^\circ;V) \to A^{*,*}(X;V \oplus L) \to 0. \end{align*}$$
-
(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 [Reference Panin and WalterPW18, §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 [Reference AnanyevskiyAna20, 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 [Reference AnanyevskiyAna20, 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 [Reference AnanyevskiyAna20, 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 [Reference AnanyevskiyAna20, 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 [Reference AnanyevskiyAna20, §4] when A is an arbitrary
$\operatorname {\mathrm {SL}}$
-oriented theory, but as explained in [Reference AnanyevskiyAna20, 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.
-
(i) There exist an isomorphism
$\Sigma ^{4,2}\mathbf {1}_S \simeq \operatorname {\mathrm {Th}}_S(L^{\oplus 2})$
in
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
. -
(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))
$$ \begin{align} \operatorname{\mathrm{Th}}(\alpha^{\oplus 2}) \colon \operatorname{\mathrm{Th}}_S(1^{\oplus 2}) \xrightarrow{\sim} \operatorname{\mathrm{Th}}_S(L^{\oplus 2}). \end{align} $$
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
$\operatorname {\mathrm {SH}}(S)$
$$ \begin{align} \operatorname{\mathrm{Th}}(\operatorname{\mathrm{id}}_{L^{\otimes s_1}} \otimes \alpha^{\otimes s_2-s_1}) \colon \operatorname{\mathrm{Th}}_S(L^{\otimes s_1}) \xrightarrow{\sim} \operatorname{\mathrm{Th}}_S(L^{\otimes s_2}). \end{align} $$
(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
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
.
4 Nowhere-vanishing sections of odd rank bundles
4.1 Projective bundles
The results of this section are slight generalisations of those of [Reference AnanyevskiyAna16a, §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
$\operatorname {\mathrm {Spt}}(S)$
$$ \begin{align} \Sigma^{2r,r} \mathbf{1}_S = \operatorname{\mathrm{Th}}_S(1^{\oplus r}) \simeq \operatorname{\mathrm{Th}}_{\mathbb{P}^0}(\mathcal{O}(\mathcal{D})) \xrightarrow{\iota_k} \operatorname{\mathrm{Th}}_{\mathbb{P}^k}(\mathcal{O}(\mathcal{D})). \end{align} $$
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).
-
(i) If k and d are odd, then
$\operatorname {\mathrm {Th}}_{\mathbb {P}^k}(\mathcal {O}(\mathcal {D})) = 0$
in
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
. -
(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
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
. -
(iii) If k is even and d is odd, then
$\operatorname {\mathrm {Th}}_{\mathbb {P}^k}(\mathcal {O}(\mathcal {D})) \simeq \Sigma ^{2(k+r),k+r}\mathbf {1}_S$
. -
(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
$\operatorname {\mathrm {SH}}(S)$
$$ \begin{align} \operatorname{\mathrm{Th}}_{\mathbb{P}^{k-1}}(\mathcal{O}(\mathcal{D})) \to \operatorname{\mathrm{Th}}_{\mathbb{P}^k}(\mathcal{O}(\mathcal{D})) \to \Sigma^{2(k+r),k+r}\mathbf{1}_S \to \Sigma^{1,0} \operatorname{\mathrm{Th}}_{\mathbb{P}^{k-1}}(\mathcal{O}(\mathcal{D})). \end{align} $$
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
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
.
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
$\operatorname {\mathrm {Spt}}(S)$
which induces a morphism of the homotopy cofibers of the horizontal morphisms:
$$ \begin{align} \operatorname{\mathrm{Th}}_{\mathbb{P}(E)}(\mathcal{O}(-1)) \to \operatorname{\mathrm{Th}}_S(E) \in \operatorname{\mathrm{SH}}(S). \end{align} $$
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 $
$$ \begin{align} 0 \to Q \to p^*E \to 1 \to 0. \end{align} $$
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., [Reference RiouRio10, 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
$\operatorname {\mathrm {SH}}(S)$
$$ \begin{align} \operatorname{\mathrm{Th}}(\operatorname{\mathrm{id}}_V \otimes \alpha^{\otimes s}) \colon \operatorname{\mathrm{Th}}_S(V) \to \operatorname{\mathrm{Th}}_S(V \otimes L^{\otimes s}). \end{align} $$
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 [Reference Morel and VoevodskyMV99, §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 [Reference Morel and VoevodskyMV99, 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 [Reference Morel and VoevodskyMV99, 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 [Reference Morel and VoevodskyMV99]), 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., [Reference HirschhornHir03, 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 [Reference Morel and VoevodskyMV99, 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 [Reference Morel and VoevodskyMV99, 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 [Reference Morel and VoevodskyMV99, 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 [Reference Morel and VoevodskyMV99, 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 [Reference Morel and VoevodskyMV99, 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 [Reference Morel and VoevodskyMV99, 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 [Reference Morel and VoevodskyMV99, 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 [Reference Morel and VoevodskyMV99, 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
$$ \begin{align} B_m(G \times G') = B_mG \times B_mG'. \end{align} $$
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
$B_mG$
:
$$ \begin{align} C_m(\chi)= (E_mG \times {\mathbb{A}}^1)/G. \end{align} $$
The assignment
$\chi \mapsto C_m(\chi )$
satisfies
$$ \begin{align} C_m(\chi \chi') = C_m(\chi) \otimes C_m(\chi'), \end{align} $$
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
$$ \begin{align} C_m(\chi) \times_{B_mG} B_mH \simeq C_m(\chi|_H). \end{align} $$
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
$$ \begin{align} C_m(\tilde{\chi}) = C_m(\chi) \times B_mG'. \end{align} $$
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
$$ \begin{align} 1 \to H \to G \xrightarrow{\chi} \mathbb{G}_m \to 1. \end{align} $$
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
$m\in \mathbb {N}\smallsetminus \{0\}$
$$ \begin{align} (B_mH)_+ \xrightarrow{{p_m}_+} (B_mG)_+ \to \operatorname{\mathrm{Th}}_{B_mG}(C_m(\chi)). \end{align} $$
More generally (as in (1.4)), if
$V \to B_mG$
is a vector bundle, we have a cofiber sequence in
${\mathrm{Spc}_{\bullet }}(S)$
,
$$ \begin{align} \operatorname{\mathrm{Th}}_{B_mH}(V) \to \operatorname{\mathrm{Th}}_{B_mG}(V) \to \operatorname{\mathrm{Th}}_{B_mG}(C_m(\chi) \oplus V). \end{align} $$
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
${\mathrm{Spc}_{\bullet }}(S)$
$$ \begin{align} (\mathrm{B} H)_+ \to (\mathrm{B} G)_+ \to \operatorname{\mathrm{Th}}_{\mathrm{B} G}(C(\chi)). \end{align} $$
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
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
5.3.5
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
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
$$ \begin{align} \Sigma^{\infty}_+ \mathrm{B} H \simeq \Sigma^{\infty}_+ \mathrm{B} G \oplus \Sigma^{-2,-1}\operatorname{\mathrm{Th}}_{\mathrm{B} G}(C(\chi)) \in \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align} $$
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
$\operatorname {\mathrm {Spt}}(S)$
$$ \begin{align} \Sigma^{2,1} \mathbf{1}_S \to \operatorname{\mathrm{Th}}_{\mathbb{P}^\infty}(\mathcal{O}(n)). \end{align} $$
Proposition 6.1.1. Let
$n\in \mathbb {Z}$
. The following hold in
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
:
-
(i) The morphism
$\iota _\infty $
induces an isomorphism
$\mathbf {1}_S \simeq \Sigma ^{\infty }_+\mathbb {P}^\infty $
. -
(ii) If n is odd, then
$\operatorname {\mathrm {Th}}_{\mathbb {P}^\infty }(\mathcal {O}(n)) = 0$
. -
(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
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
:
-
(i) The natural morphism
$\mathbf {1}_S \to \Sigma ^{\infty }_+\mathrm {B}\mathbb {G}_m$
is an isomorphism. -
(ii) If n is odd, the natural morphism
$\mathbf {1}_S \to \Sigma ^{\infty }_+\mathrm {B}\mu _n$
is an isomorphism. -
(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
$n\colon \mathbb {G}_m \to \mathbb {G}_m$
.
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 [Reference Panin and WalterPW18, §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
$$ \begin{align} \operatorname{\mathrm{Th}}_{B_mP_n}(C_m(\nu_n) \oplus C_m(q_n))=0= \operatorname{\mathrm{Th}}_{B_mP_n}(C_m(q_n)) \in \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align} $$
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 [Reference AnanyevskiyAna15, 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 [Reference AnanyevskiyAna15, 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 [Reference AnanyevskiyAna15], the problem being with [Reference AnanyevskiyAna15, Lemma 6] (which is used to prove [Reference AnanyevskiyAna15, 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
$\operatorname {\mathrm {Spc}}(S)$
$$ \begin{align} \operatorname{\mathrm{BGL}}_n \to \operatorname{\mathrm{BGL}}_{n+1}. \end{align} $$
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
$$ \begin{align} (1^{\oplus m})^{\oplus n} =(1^{\oplus m})^{\oplus n} \oplus 0 \subset (1^{\oplus m})^{\oplus n} \oplus 1^{\oplus m} = (1^{\oplus m})^{\oplus n+1}. \end{align} $$
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
$$ \begin{align} \Sigma^{\infty}_+ \operatorname{\mathrm{BSL}}_n = \Sigma^{\infty}_+ \operatorname{\mathrm{BGL}}_n \oplus \Sigma^{-2,-1}\operatorname{\mathrm{Th}}_{\operatorname{\mathrm{BGL}}_n}(C({\det}_n)) \in \operatorname{\mathrm{SH}}(S)[\eta^{-1}]. \end{align} $$
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
$B_m\operatorname {\mathrm {GL}}_n$
$$ \begin{align} \det \mathcal{U}_n \simeq C_m({\det}_n). \end{align} $$
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
$$ \begin{align} p^*(\det \mathcal{U}_{2r}) \otimes \mathcal{O}_{\mathbb{P}(\mathcal{Q}_{2r})}(-1) \simeq q^*(\det \mathcal{U}_{2r+1}) \simeq q^*C_m({\det}_{2r+1}). \end{align} $$
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 [Reference LevineLev19, 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 [Reference LevineLev19, Theorem 4.1].)
Appendix A An invariant of
$\mu _2$
-torsors
We have proved in (6.1.3) that in
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
, for
$n\in \mathbb {N}$
$$\begin{align*}\Sigma^{\infty}_+ \mathrm{B} \mu_n = \begin{cases} \mathbf{1}_S & \text{if } n=0 (\text{i.e., } \mu_n = \mathbb{G}_m) \text{ or } n \text{ is odd,}\\ \Sigma^{\infty}_+ \mathbb{G}_m = \mathbf{1}_S \oplus \mathbf{1}_S & \text{if } n>0 \text{ is even.} \end{cases} \end{align*}$$
Thus, when
$n>0$
is even, there is essentially one nontrivial invariant of
$\mu _n$
-torsors over S in
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
, in the form of an element of
$\operatorname {\mathrm {End}}_{\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]}(\mathbf {1}_S)$
. Moreover, it also follows from (6.1.3) that the morphism
$\mathrm {B} \mu _2 \to \mathrm {B} \mu _{2r}$
becomes an isomorphism in
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
for
$r>0$
. Therefore, the above-mentioned invariant of
$\mu _n$
-torsors is induced by an invariant of
$\mu _2$
-torsors, which is, however, not really explicit from this description. In this section, we provide an explicit construction of this invariant (the connection with the above discussion is made in (A.9) below).
A.1
Let
$L\to S$
be a line bundle. As observed by Ananyevskiy [Reference AnanyevskiyAna20, Lemma 4.1], the isomorphism of S-schemes
$L^\circ \xrightarrow {\sim } (L^\vee )^\circ $
, given locally by
$l \mapsto l^\vee $
, where
$l^\vee (l)=1$
, induces an isomorphism in
$H_{\bullet }(S)$
$$\begin{align*}\sigma_L \colon \operatorname{\mathrm{Th}}_S(L) \xrightarrow{\sim} \operatorname{\mathrm{Th}}_S(L^\vee). \end{align*}$$
A.2
If
$\varphi \colon L \to M$
is an isomorphism of line bundles over X, we have (see (1.5))
$$ \begin{align} \sigma_L = \operatorname{\mathrm{Th}}(\varphi^\vee) \circ \sigma_M \circ \operatorname{\mathrm{Th}}(\varphi). \end{align} $$
Definition A.3. It will be convenient to think of a
$\mu _2$
-torsor over S as a pair
$(L,\lambda )$
, where
$L\to S$
is a line bundle, and
$\lambda \colon L \xrightarrow {\sim } L^\vee $
is an isomorphism of line bundles over S. Isomorphisms
$(L,\lambda ) \to (L',\lambda ')$
are given by isomorphisms of line bundles
$\varphi \colon L \xrightarrow {\sim } L'$
such that
$\lambda = \varphi ^\vee \circ \lambda ' \circ \varphi $
. The set of isomorphism classes of
$\mu _2$
-torsors is denoted
$H^1_{et}(S,\mu _2)$
; it is endowed with a group structure induced by the tensor product of line bundles.
Definition A.4. Consider a
$\mu _2$
-torsor, given by a line bundle
$L\to S$
and an isomorphism
$\lambda \colon L\xrightarrow {\sim } L^\vee $
. Let us consider the composite isomorphism in
$\operatorname {\mathrm {SH}}(S)$
(see (A.1) for the definition of
$\sigma _L$
, and (1.5) for that of
$\operatorname {\mathrm {Th}}(\lambda )$
)
$$\begin{align*}\operatorname{\mathrm{Th}}_S(L) \xrightarrow{\operatorname{\mathrm{Th}}(\lambda)} \operatorname{\mathrm{Th}}_S(L^\vee) \xrightarrow{\sigma_L^{-1}} \operatorname{\mathrm{Th}}_S(L). \end{align*}$$
This yields an element
$a_{(L,\lambda )} = \Sigma ^{-L}(\sigma _L^{-1} \circ \operatorname {\mathrm {Th}}(\lambda )) \in \operatorname {\mathrm {Aut}}_{\operatorname {\mathrm {SH}}(S)}(\mathbf {1}_S)$
. We define
$$\begin{align*}\alpha(L,\lambda) = (a_{(1,\operatorname{\mathrm{can}})})^{-1} \circ a_{(L,\lambda)} \in \operatorname{\mathrm{Aut}}_{\operatorname{\mathrm{SH}}(S)}(\mathbf{1}_S), \end{align*}$$
where
$\operatorname {\mathrm {can}} \colon 1 \to 1^\vee $
is the canonical isomorphism of line bundles over S. (The element
$a_{(1,\operatorname {\mathrm {can}})}$
corresponds to the element
$\epsilon $
of [Reference MorelMor04, §6.1].)
A.5
This construction is compatible with pullbacks, in the sense that if
$f\colon R \to S$
is a morphism of noetherian schemes of finite dimension and
$(L,\lambda )$
a
$\mu _2$
-torsor, then the composite in
$\operatorname {\mathrm {SH}}(R)$
$$\begin{align*}\mathbf{1}_R = f^*\mathbf{1}_S \xrightarrow{f^*\alpha(L,\lambda)} f^*\mathbf{1}_S =\mathbf{1}_R \end{align*}$$
is
$\alpha (f^*L,f^*\lambda )$
.
Example A.6. Consider a
$\mu _2$
-torsor
$(L,\lambda )$
, where
$L=1$
is the trivial line bundle. Then
$\lambda = u \operatorname {\mathrm {can}}$
for some
$u \in H^0(S,\mathbb {G}_m)$
, hence
$\operatorname {\mathrm {Th}}(\lambda ) = \operatorname {\mathrm {Th}}(\operatorname {\mathrm {can}}) \circ \operatorname {\mathrm {Th}}(u\operatorname {\mathrm {id}}_1)$
. Therefore, (see (1.6))
$$ \begin{align*} \alpha(1,\lambda) &= \Sigma^{-2,-1}((\sigma_1^{-1} \circ \operatorname{\mathrm{Th}}(\operatorname{\mathrm{can}}))^{-1} \circ (\sigma_1^{-1} \circ \operatorname{\mathrm{Th}}(\lambda)) \\ &= \Sigma^{-2,-1}(\operatorname{\mathrm{Th}}(\operatorname{\mathrm{can}})^{-1} \circ \operatorname{\mathrm{Th}}(\lambda))\\ &= \Sigma^{-2,-1} \operatorname{\mathrm{Th}}(u\operatorname{\mathrm{id}}_1)\\ &= \langle u\rangle. \end{align*} $$
Proposition A.7. The assignment
$(L,\lambda ) \mapsto \alpha (L,\lambda )$
induces a morphism of pointed sets
$$\begin{align*}\alpha \colon H^1_{et}(S,\mu_2) \to \operatorname{\mathrm{Aut}}_{\operatorname{\mathrm{SH}}(S)}(\mathbf{1}_S). \end{align*}$$
Proof. By construction, we have
$\alpha (1,\operatorname {\mathrm {can}})=\operatorname {\mathrm {id}}$
. Consider an isomorphism of
$\mu _2$
-torsors
$(L,\lambda ) \xrightarrow {\sim } (M,\mu )$
(see (A.3)), given by an isomorphism
$\varphi \colon L \xrightarrow {\sim } M$
. Let us set
$a_L = a_{(L,\lambda )}$
and
$a_M = a_{(M,\mu )}$
(see (A.4)). Then, in
$\operatorname {\mathrm {Aut}}_{\operatorname {\mathrm {SH}}(S)}(\operatorname {\mathrm {Th}}_S(L))$
,
$$ \begin{align*} \Sigma^La_L = \sigma_L^{-1} \circ \operatorname{\mathrm{Th}}(\lambda) &= \sigma_L^{-1} \circ \operatorname{\mathrm{Th}}(\varphi^\vee) \circ \operatorname{\mathrm{Th}}(\mu) \circ \operatorname{\mathrm{Th}}(\varphi) && \text{(as } \lambda = \varphi^\vee \circ \mu \circ \varphi) \\ &= \operatorname{\mathrm{Th}}(\varphi)^{-1} \circ \sigma_M^{-1} \circ \operatorname{\mathrm{Th}}(\mu) \circ \operatorname{\mathrm{Th}}(\varphi) && \text{by equation (A.2.a)}\\ &= \operatorname{\mathrm{Th}}(\varphi)^{-1} \circ (\Sigma^Ma_M) \circ \operatorname{\mathrm{Th}}(\varphi) \\ &= \operatorname{\mathrm{Th}}(\varphi)^{-1} \circ \operatorname{\mathrm{Th}}(\varphi) \circ (\Sigma^L a_M) && \text{(by equation (1.5.b))}\\ &= \Sigma^La_M, \end{align*} $$
whence
$a_L = a_M$
, and
$\alpha (L,\lambda ) = \alpha (M,\mu )$
.
Proposition A.8. The assignment
$(L,\lambda ) \mapsto \alpha (L,\lambda )$
induces a group morphism
$$\begin{align*}\alpha \colon H^1_{et}(S,\mu_2) \to \operatorname{\mathrm{Aut}}_{\operatorname{\mathrm{SH}}(S)[\eta^{-1}]}(\mathbf{1}_S). \end{align*}$$
Proof. Consider
$\mu _2$
-torsors given by line bundles
$L,M$
over S and isomorphisms
$L\xrightarrow {\sim } L^\vee $
and
$M \xrightarrow {\sim } M^\vee $
. As the functor
$f^* \colon \operatorname {\mathrm {SH}}(S)[\eta ^{-1}] \to \operatorname {\mathrm {SH}}(L^\circ )[\eta ^{-1}]$
is faithful (2.2.8), by functoriality (A.5) we may assume that the line bundle L is trivial. Similarly, we may assume that M is also trivial. By (A.7), we may assume that
$L=M=1$
. Then, in view of (A.6), the statement follows from equation (1.6.a).
Remark A.9. Let
$\rho \colon \mathbb {G}_m \to \mathrm {B} \mu _2$
be the map classifying the
$\mu _2$
-torsor
$(1,t\operatorname {\mathrm {id}})$
over
$\mathbb {G}_m$
, where
$t\in H^0(\mathbb {G}_m,\mathbb {G}_m)$
is the tautological section. Recall from (6.1.3.iii) that
$\Sigma ^{\infty }_+ \rho \colon \Sigma ^{\infty }_+ \mathbb {G}_m \to \Sigma ^{\infty }_+ \mathrm {B} \mu _2$
is an isomorphism in
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
. If a
$\mu _2$
-torsor
$(L,\lambda )$
is classified by the map
$f \colon S \to \mathrm {B} \mu _2$
, we claim that
$\alpha (L,\lambda )$
is the composite in
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]$
(the map
$\pi _1$
was defined in (2.2.1))
$$ \begin{align} \mathbf{1}_S \xrightarrow{\Sigma^{\infty}_+ f} \Sigma^{\infty}_+ \mathrm{B} \mu_2 \xrightarrow{(\Sigma^{\infty}_+ \rho)^{-1}} \Sigma^{\infty}_+ \mathbb{G}_m \xrightarrow{\Sigma^{-2,-1}\Sigma^{\infty}_+ \pi_1} \mathbf{1}_S. \end{align} $$
Indeed, applying the functor
$\operatorname {\mathrm {SH}}(S)[\eta ^{-1}] \to \operatorname {\mathrm {SH}}(L^\circ )[\eta ^{-1}]$
which is faithful by (2.2.8) and using (A.7), we may assume that
$L=1$
. Then
$\lambda $
is given by multiplication by an element
$u \in H^0(S,\mathbb {G}_m)$
, and the map f factors as
$S \xrightarrow {u} \mathbb {G}_m \xrightarrow {\rho } \mathrm {B} \mu _2$
. Denoting by
$p \colon \mathbb {G}_m \to S$
is the projection, the composite (A.9.a) is given by
$$\begin{align*}\mathbf{1}_S \xrightarrow{\Sigma^{\infty}_+ u} \Sigma^{\infty}_+ \mathbb{G}_m \xrightarrow{\langle t \rangle} \Sigma^{\infty}_+ \mathbb{G}_m \xrightarrow{\Sigma^{\infty}_+ p} \mathbf{1}_S, \end{align*}$$
which coincides with
$\langle u \rangle \in \operatorname {\mathrm {Aut}}_{\operatorname {\mathrm {SH}}(S)[\eta ^{-1}]}(S)$
. Thus, the claim follows from (A.6).
Acknowledgments
I am grateful to the referee for his/her suggestions and in particular for noticing that the coequaliser diagram of (2.2.7) is split.
This work was supported by the DFG research grant HA 7702/5-1 and Heisenberg grant HA 7702/4-1.
Competing interests
The authors have no competing interest to declare.
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