Odd rank vector bundles in eta-periodic motivic homotopy theory

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


Introduction
Around forty years ago, Arason computed the Witt groups of projective spaces [Ara80]. This computation was later revisited by Gille [Gil01], Walter [Wal03], and Nenashev [Nen09]. It exhibited Witt groups as a somewhat exotic cohomology theory, whose value on projective spaces differs quite drastically from what is obtained in more classical cohomology theories such as Chow groups or K-theory. It is now understood that this behaviour reflects the lack of GL-orientation in Witt theory.
Ananyevskiy observed in [Ana16a] that the key property of Witt groups permitting to perform these computations turns out to be the fact that the Hopf map η : A 2 t0u Ñ P 1 , px : yq Þ Ñ rx : ys induces by pullback an isomorphism of Witt groups WpP 1 q " Ý Ñ WpA 2 t0uq. He thus extended in [Ana16a] the above-mentioned computations to arbitrary cohomology theories in which η induces an isomorphism.
Inverting the Hopf map η in the motivic stable homotopy category SHpSq over a base scheme S yields its η-periodic version SHpSqrη´1s, a category which has been studied in details by Bachmann-Hopkins [BH20]. In this paper, we lift Ananyevskiy's computations of the cohomology of projective spaces to the η-periodic stable homotopy category: we obtain for instance that a projective bundle of even relative dimension becomes an isomorphism in SHpSqrη´1s.
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 SL c -orientability of Witt groups (see below). We show that that property of Witt groups in fact follows from their η-periodicity alone (see (4.3.2) for a more general statement): Proposition. Let V Ñ X be a vector bundle and L Ñ X a line bundle. Then we have an isomorphism of Thom spaces Th X pV q ≃ Th X pV b L b2 q ∈ SHpSqrη´1s.
Panin and Walter introduced [PW18, §3] the notion of SL c -orientability for algebraic cohomology theories, which consists in the data of Thom classes for vector bundles equipped with a square root of their determinant, and proved that Hermitian K-theory is SL c -oriented. Ananyevskiy later showed [Ana20, Theorem 1.2] that a cohomology theory is SL c -oriented as soon as it is a Zariski sheaf in bidegree p0, 0q, and pointed out [Ana20, Theorem 1.1] the close relations between SL c -orientations and 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 η-periodic context: Theorem. Every SL-orientation of an η-periodic motivic commutative ring spectrum is induced by a unique 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˝the complement of the zero-section in E.
(i) The projection E˝Ñ X admits a section in SHpSqrη´1s.
(ii) The diagram E˝ˆX E˝Ñ E˝Ñ X becomes a split coequaliser diagram in SHpSqrη´1s.
The first assertion may be viewed as a splitting principle, while the second permits to perform a form of descent. To some extent, this proposition allows us to assume that odd rank vector bundle admit a nowhere-vanishing section (once η is inverted); in particular that line bundles are trivial.
Finally, we provide applications to the computation in SHpSqrη´1s of the étale classifying spaces of certain algebraic groups.
From this theorem, we deduce a computation in SHpSqrη´1s of the classifying space of an arbitrary diagonalisable group. We obtain that all invariants of torsors under a diagonalisable group with values in an η-periodic cohomology theory arise from a single invariant of µ 2 -torsors. In the appendix, we present an explicit construction of that invariant, exploiting the identification of the group µ 2 with the orthogonal group O 1 .
The first result can be viewed as a companion of the theorem on orientations stated above, and cements the idea that the groups SL c n and SL n are the same in the eyes of η-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 η-periodic stable homotopy theory.
The morphism BSL 2r Ñ BGL 2r is not an isomorphism in SHpSqrη´1s, but we show that it admits a section, expressing the fact that every invariant (with values in an η-periodic cohomology theory) of even-dimensional vector bundles is determined by its value on those bundles having trivial determinant.
Finally, let us mention that the results of this paper serve as a starting point for the paper [Hau22] on Pontryagin classes.

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 Sm S . All schemes will be implicitly assumed to belong to Sm S , and the notation A n , P n , G m will refer to the corresponding S-schemes. We will denote by 1 the trivial line bundle over a given scheme in Sm S . 1.2. We will use the A 1 -homotopy theory introduced by Morel-Voevodsky [MV99]. We will denote by SpcpSq the category of motivic spaces (i.e. simplicial presheaves on Sm S ), by Spc ‚ pSq its pointed version, and by SptpSq the category of T -spectra, where T " A 1 {G m . We endow these with the motivic equivalences, resp. stable motivic equivalences, and denote by HpSq, H ‚ pSq, SHpSq the respective homotopy categories. We refer to e.g. [PPR09, Appendix A] for more details.
The spheres are denoted as usual by S p,q ∈ Spc ‚ pSq for p, q ∈ N with p ě q (where T ≃ S 2,1 ). The motivic sphere spectrum Σ ∞ S will be denoted by 1 S ∈ SptpSq. When A is a motivic spectrum, we denote its pp, qq-th suspension by Σ p,q A " S p,q^A . This yields functors Σ p,q : SHpSq Ñ SHpSq for p, q ∈ Z.
1.3. When E Ñ X is a vector bundle with X ∈ Sm S , we denote by E˝" E X the complement of the zero-section. The Thom space of E is the pointed motivic space Th X pEq " E{E˝. We will write Th X pEq ∈ SptpSq instead of Σ ∞ Th X pEq, in order to lighten the notation. When g : Y Ñ X is a morphism in Sm S , we will usually write Th Y pEq instead of Th Y pg˚Eq. Since E Ñ X is a weak equivalence, we have a cofiber sequence in Spc ‚ pSq, where p : E˝Ñ X is the projection, pE˝q`pÝ Ñ X`Ñ Th X pEq.
If F Ñ S is a vector bundle and f : X Ñ S the structural morphism, we have by [MV99, Proposition 3.2.17 (1)] a natural identification in Spc ‚ pSq When V Ñ S is a vector bundle, we denote by Σ V : SHpSq Ñ SHpSq the derived functor induced by A Þ Ñ A^Th S pV q. It is an equivalence of categories, with inverse denoted by Σ´V .
More generally, if V Ñ X is a vector bundle, we have a cofiber sequence in Spc ‚ pSq This may be deduced from (1.4.a) by first reducing to the case X " S using the functor f 7 of (1.9) below, and then applying the functor´^Th X pV q (both of which preserve homotopy colimits), in view of (1.3.b).
1.5. Let ϕ : E " Ý Ñ F be an isomorphism of vector bundles over X ∈ Sm S . Then ϕ induces a weak equivalence in Spc ‚ pSq (and SptpSq) Thpϕq : Th X pEq Ñ Th X pF q.
If X " S and f ∈ End SHpSq p1 S q, then we have in SHpSq (This follows from the fact that, as morphisms 1 S^T h S pEq Ñ 1 S^T h S pF q pf^id Th S pF q q˝pid 1 S^T hpϕqq " f^Thpϕq " pid 1 S^T hpϕqq˝pf^id Th S pEq q.q 1.6. (See [Mor04, Lemma 6.3.4].) Let X ∈ Sm S and u ∈ H 0 pX, G m q. Consider the automorphism u id 1 : 1 Ñ 1 of the trivial line bundle over X, and set xuy " Σ´2 ,´1 Thpu id 1 q ∈ Aut SHpSq pΣ ∞ Xq.
When X " S and A ∈ SHpSq, we will denote again by xuy ∈ Aut SHpSq pAq the morphism If f : A Ñ B is a morphism in SHpSq, then (1.6.b) f˝xuy " xuy˝f.
1.8. We will consider the categories Spc ‚ pSqrη´1s and SptpSqrη´1s obtained by monoidally inverting the map η of (1.7), which can be constructed as left Bousfield localisations, as discussed in [Bac18,§6]. Their respective homotopy categories will be denoted by H ‚ pSqrη´1s and SHpSqrη´1s, and we will usually omit the mention of the localisation functors.
A spectrum A ∈ SptpSq is called η-periodic if the map is an isomorphism in SHpSq. The full subcategory of such objects in SptpSq can be identified SptpSqrη´1s.
1.9. Let X ∈ Sm S with structural morphism f : X Ñ S. Then there are Quillen adjunctions The functor f˚is induced by base-change, while f 7 arises from viewing a smooth Xscheme as a smooth S-scheme by composing with f . These induce Quillen adjunctions f 7 : Spc ‚ pSqrη´1s Ô Spc ‚ pSqrη´1s : f˚; f 7 : SptpXqrη´1s Ô SptpSqrη´1s : f˚.
We will also use the notation f˚, f 7 for the derived functors on the respective homotopy categories.
1.10. (See e.g. [DHI04].) Let V Ñ X be a vector bundle with X ∈ Sm S , and U α an open covering of X. Then the map Th Uα pV | Uα q¯Ñ Th X pV q is a weak equivalence.
2. Splitting G m -torsors 2.1. Local splitting. In this section we consider the schemes A 1 , G m , P 1 , A 2 t0u as pointed motivic spaces, respectively via 1, 1, r1 : 1s, p1, 1q. We recall that T " A 1 {G m . We have a chain of weak equivalences where the first arrow is induced by the immersion A 1 Ñ P 1 , x Þ Ñ rx : 1s, and the quotient P 1 { A 1 is taken with respect to the immersion A 1 Ñ P 1 , y Þ Ñ r1 : ys.
We first recall a well-known fact (see e.g. [Ana20, Lemma 6.2] for a stable version): 2.1.1. Lemma. Let u ∈ H 0 pS, G m q. Then the morphism Thpu 2 id 1 q : T Ñ T (see (1.5)) coincides with the identity in H ‚ pSq.
Proof. The endomorphism ϕ : P 1 Ñ P 1 given by rx : ys Þ Ñ ru 2 x : ys " rux : u´1ys is induced by the matrix s a product of transvections, the endomorphism ϕ induces the identity endomorphism of P 1 in H ‚ pSq (see e.g. [Ana16a, Lemma 1]). The map ϕ stabilises the copies of A 1 given by x Þ Ñ rx : 1s and y Þ Ñ r1 : ys, and restricts to u 2 id 1 on the former. Thus the statement follows from the isomorphism (2.1.0.a).

Excision yields isomorphisms in
omposing with the quotient A 2 t0u Ñ pA 2 t0uq{pA 1ˆG m q, this yields a map (2.1.
2.1.3. Lemma. The projection p : G m Ñ S induces a cofiber sequence in Spc ‚ pSq Proof. This follows from the consideration of the following commutative diagram in Spc ‚ pSq, whose rows are cofiber sequences and where the curved arrow is the weak equivalence induced by the projection G mÂ 1 Ñ G m .
where t ∈ H 0 pG m , G m q is the tautological section and p : G m Ñ S the projection (and T^pG m q`is identified with Th Gm p1q).

Proof. Consider the commutative diagram in Sm
where the upper horizontal arrow is the natural open immersion, the lower horizontal arrow is given by x Þ Ñ rx : 1s, and µ is given by px, yq Þ Ñ xy´1. Excision yields the isomorphisms in the commutative diagram in H ‚ pSq whereμ is induced by µ, and the lower horizontal arrows are the morphisms of (2. here the left inner square is homotopy cocartesian. Applying (2.1.1) over the base G m and using the functor p 7 of (1.9), we have in H ‚ pSq Thpt id 1 q " Thpt´1 id 1 q : T^pG m q`Ñ T^pG m q`.  Ý Ñ p˚L. We define a morphism in H ‚ pSq (recall that T " Th S p1q, and see (1.5)) 2.2.2. Example. Assume that L " 1. Then the tautological trivialisation τ : 1 Ñ 1 of the trivial line bundle over L˝" G m is given by multiplication by the tautological section t ∈ H 0 pG m , G m q, hence π 1 coincides with morphism π of (2.1.5).
2.2.3. Let L Ñ S be a line bundle. If f : R Ñ S is a scheme morphism, then the functor f˚: Spc ‚ pSq Ñ Spc ‚ pRq maps π L to π f˚L .
By (1.10) and (1.9) (and in view of (2.2.3)), the fact that F ≃˚may be verified Zariskilocally on S. We may thus assume that L is trivial. By (2.2.4) we may further assume that L " 1, so that L˝" G m . Then, in view of (2.2.2) the result follows from (2.1.5) The next statement was initially inspired by [Lev19, Proof of Theorem 4.1]: 2.2.6. Corollary. Let L Ñ S be a line bundle. Then in the notation of (2.2.1), we have an isomorphism in SHpSqrη´1s Proof. The square induced in SptpSqrη´1s by the square of (2.2.5) is homotopy cocartesian, hence also homotopy cartesian (see e.g. [Hov99, Remark 7.1.12]). This yields an isomorphism from which the result follows by applying the functor Σ´2 ,´1 .
2.2.7. Corollary. Let L Ñ S be a line bundle, and V Ñ S a vector bundle.
(i) The natural map Th L˝p V q Ñ Th S pV q extends to a natural isomorphism in SHpSqrη´1s (ii) Denote by p : L˝Ñ S and p 1 , p 2 : L˝ˆS L˝Ñ L˝the projections. Then Proof. Statement (i) follows by applying the auto-equivalence Σ V : SHpSqrη´1s Ñ SHpSqrη´1s to the decomposition of (2.2.6), in view of (1.3.b).
Certainly in the diagram of (ii) we have p˝p 1 " p˝p 2 . The isomorphism (i) yields a section s : On the other hand, in the commutative diagram in SHpSqrη´1s the upper composite is t, while the lower one is s. Therefore p 2˝t " s˝p as endomorphisms of Th L˝p V q in SHpSqrη´1s, proving (ii).
2.2.8. Corollary. Let L Ñ S be a line bundle, and denote by p : L˝Ñ S the projection. Then the functor p˚: SHpSqrη´1s Ñ SHpL˝qrη´1s is faithful and conservative.
Proof. By the smooth projection formula and (2.2.6), the composite p 7˝p˚: SHpSqrη´1s Ñ SHpSqrη´1s decomposes as which is faithful, hence so is p˚. The above formula also shows that p 7˝p˚r eflects zero-objects, hence so does p˚. Since p˚is triangulated, it is conservative.
2.2.9. Remark. The results of this section on line bundles will be generalised to odd rank vector bundles in §4.2.

Applications to twisted cohomology
3.1. Cohomology theories represented by ring spectra.
3.1.1. Let A ∈ SptpSq be a motivic spectrum. For a pointed motivic space X we write and A˚,˚pX q " À p,q∈Z A p,q pX q. When X is a smooth S-scheme, we will write A˚,˚pXq instead of A˚,˚pX`q. If E Ñ X is a vector bundle of constant rank r, we write A p,q pX; Eq " A p`2r,q`r pTh X pEqq, and extend this notation to arbitrary vector bundles in an obvious way. A morphism f : Y Ñ X of pointed motivic spaces (resp. of smooth S-schemes) induces a pullback f˚: A˚,˚pX q Ñ A˚,˚pYq.
3.1.2. A commutative ring spectrum will mean a commutative monoid in pSHpSq,^, 1 S q. When A ∈ SHpSq is a commutative ring spectrum and X ∈ Sm S , then A˚,˚pXq is naturally a ring, and A˚,˚pX; Eq an A˚,˚pXq-module. When u ∈ H 0 pX, G m q, we will write xuy ∈ A 0,0 pXq instead of xuy˚p1q (see (1.6)).

If
A is an η-periodic motivic spectrum, for any pointed motivic space X , we have natural isomorphisms for p, q ∈ Z A p,q pX q " Hom SHpSqrη´1s pΣ ∞ X , Σ p,q Aq.

Proposition. Let
A be an η-periodic motivic spectrum. Let X ∈ Sm S . Let L Ñ X be a line bundle, and V Ñ X a vector bundle.
(i) Denoting by p : L˝Ñ X the projection, we have a split short exact sequence (ii) Denoting by p 1 , p 2 : L˝ˆX L˝Ñ L˝the projections, we have an exact sequence Proof. This follows by applying (2.2.7) over the base X to the image of A under the pullback SptpSq Ñ SptpXq.

3.2.1.
Definition. An SL-oriented vector bundle over a scheme X is a pair pE, δq, where E Ñ X is a vector bundle and δ : 1 " Ý Ñ det E is an isomorphism of line bundles. We will also say that δ is an SL-orientation of the vector bundle E Ñ X. An isomorphism of SL-oriented vector bundles pE, δq " Ý Ñ pF, ǫq is an isomorphism of vector bundles ϕ : E " Ý Ñ F such that pdet ϕq˝δ " ǫ. [PW18,§3]). An SL c -oriented vector bundle over a scheme X is a triple pE, L, λq, where E Ñ X is a vector bundle and L Ñ X a line bundle, and λ : L b2 " Ý Ñ det E is an isomorphism. We will also say that pL, λq is an SL c -orientation of the vector bundle E Ñ X. An isomorphism of SL c -oriented vector bundles pE, L, λq " Ý Ñ pF, M, µq is an isomorphism of vector bundles ϕ : E " Ý Ñ F and an isomorphism of line bundles ψ : L " Ý Ñ M such that pdet ϕq˝λ " µ˝ψ b2 .

Observe that each SL-orientation δ of a vector bundle E induces an
3.2.4. Let pE, L, λq be an SL c -oriented vector bundle, and assume that the line bundle L is trivial. Then every trivialisation α : 1 " Ý Ñ L induces an SL-orientation of E given by Observe that the SL c -oriented vector bundle induced (in the sense of (3.2.3)) by δ α is isomorphic to pE, L, λq.
3.2.5. Consider a commutative ring spectrum A ∈ SHpSq. By a SL-, resp. SL c -, orientation of A, we will mean a normalised orientation in the sense of [Ana20, Definition 3.3]. Such data consists in Thom classes th pE,δq ∈ A˚,˚pX; Eq for each SL-oriented vector bundle pE, δq over X ∈ Sm S , resp. th pE,L,λq ∈ A˚,˚pX; Eq for each SL c -oriented vector bundle pE, L, λq over X ∈ Sm S , satisfying a series of axioms.
3.2.6. Let A ∈ SHpSq be a commutative ring spectrum. Then each SL c -orientation of A induces an SL-orientation of A, by letting the Thom class of an SL-oriented vector bundle be the Thom class of the induced SL c -oriented vector bundle, in the sense of (3.2.3).
3.2.7. Lemma. Let A ∈ SHpSq be an SL-oriented commutative ring spectrum. Let pE, L, λq be an SL c -oriented vector bundle over X ∈ Sm S , and assume that the line bundle L is trivial. Then, in the notation of (3.2.4), the Thom class th pE,δαq ∈ A˚,˚pX; Eq does not depend on the choice of the trivialisation α of L.
3.2.8. Proposition. Let A ∈ SHpSq be an η-periodic commutative ring spectrum. Then every SL-orientation of A is induced (in the sense of (3.2.6)) by a unique SL corientation.
Proof. We assume given an SL-orientation of A. Let pE, L, λq be an SL c -oriented vector bundle. Denoting by p : L˝Ñ X the projection, the line bundle p˚L over L˝admits a tautological trivialisation τ : 1 " Ý Ñ p˚L. In view of (3.2.4), this yields an SL-orientation δ τ of p˚E.
First assume given an SL c -orientation of A compatible with its SL-orientation. As observed in (3.2.4), the SL c -oriented vector bundle pp˚E, p˚L, p˚λq is isomorphic to the one induced by the SL-oriented vector bundle pp˚E, δ τ q. Thus we must have p˚th pE,L,λq " th pp˚E,p˚L,p˚λq " th pp˚E,δτ q ∈ A˚,˚pL˝; p˚Eq.
Since p˚: A˚,˚pX; Eq Ñ A˚,˚pL˝; p˚Eq is injective by (3.1.4), we obtain the uniqueness part of the statement.
We now construct an SL c -orientation of A from its SL-orientation. In the situation considered at the beginning of the proof, let p 1 , p 2 : L˝ˆX L˝Ñ L˝be the projections, and set q " p˝p 1 " p˝p 2 . The tautological trivialisation τ of p˚L yields two trivialisations p1τ and p2τ of q˚L, and thus two SL-orientations α 1 " δ p1 τ and α 2 " δ p2 τ of E. However it follows from (3.2.7) that their Thom classes coincide, so that (observe that α i " pi pδ τ q for i " 1, 2) p1 th pp˚E,δτ q " th pE,α 1 q " th pE,α 2 q " p2 th pp˚E,δτ q ∈ A˚,˚pL˝ˆX L˝; q˚Eq.
Therefore it follows from (3.1.4.ii) that the element th pp˚E,δτ q ∈ A˚,˚pL˝; Eq is the image of a unique element θ pE,L,λq ∈ A˚,˚pX; Eq.
From the fact that pE, δq Þ Ñ th pE,δq defines an SL-orientation of A, we deduce at once that pE, L, λq Þ Ñ θ pE,L,λq defines an SL c -orientation of A: indeed, each axiom of [Ana20, Definition 3.3] can be verified after pulling back along p : L˝Ñ X, since p˚: A˚,˚pX; Eq Ñ A˚,˚pL˝; p˚Eq is injective by (3.1.4).
To conclude the proof, it remains to show the SL c -orientation pE, L, λq Þ Ñ θ pE,L,λq induces the original SL-orientation of A. So let us assume that the SL c -oriented vector bundle pE, L, λq is induced by an SL-oriented vector bundle pE, δq, in the sense of (3.2.3). In particular L " 1. Then the tautological trivialisation τ : 1 " Ý Ñ p˚L and the trivialisation 1 " p˚1 " p˚L yield two SL-orientations of p˚E. Their Thom classes in A˚,˚pL˝; p˚Eq coincide by (3.2.7), and they are respectively p˚θ pE,L,λq and p˚th pE,δq . Since p˚: A˚,˚pX; Eq Ñ A˚,˚pL˝; p˚Eq is injective by (3.1.4), we have θ pE,L,λq " th pE,δq ∈ A˚,˚pX; Eq, as required.
3.2.9. Remark. Ananyevskiy constructed "Thom isomorphisms" associated with SL cbundles in [Ana20, §4] when A is an arbitrary SL-oriented theory, but as explained in [Ana20, Remark 4.4] it is not clear whether this yields an SL c -orientation, the problem being the multiplicativity axiom. When A is η-periodic, our construction leads to the same Thom isomorphisms for 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 SLorientation). Thus the Thom isomorphisms constructed by Ananyevskiy do give rise to an SL c -orientation when A is η-periodic.

Twisting by doubles and squares of line bundles.
3.3.1. Proposition. Let L be a line bundle over S.
(ii) If s 1 , s 2 ∈ Z are of the same parity, there exist an isomorphism Proof. Let us first assume that the line bundle L Ñ S admits a trivialisation α : 1 " Ý Ñ L. Then we have an isomorphism in SHpSq (see (1.5)) Every trivialisation of L is of the form uα with u ∈ H 0 pS, G m q. As automorphisms of Th S p1 '2 q in SHpSq we have, Thppu id 1 q '2 q " Thppu 2 id 1 q ' id 1 q˝Thppu´1 id 1 q ' pu id 1 qq " Thppu 2 id 1 q ' id 1 q because Thppu´1 id 1 q ' pu id 1 qq is the identity of Th S p1 '2 q, being given by a product transvections (see e.g. [Ana16a, Lemma 1]). Now by (2.1.1), under the identification Th S p1 '2 q " Th S p1q^Th S p1q (see (1.3.b)), we have Therefore Thppu id 1 q '2 q is the identity of Th S p1 '2 q in SHpSq, hence so that the isomorphism Thpα '2 q in SHpSq considered in (3.3.1.a) is independent of the choice of the trivialisation α. Next let us consider the case (ii). If α : 1 " Ý Ñ L is a trivialisation, we have an isomorphism in SHpSq Thpid L bs 1 bα bs 2´s1 q : Th S pL bs 1 q " Ý Ñ Th S pL bs 2 q.
(Here and below, for r ∈ N, the notation α b´r refers to the morphism ppα _ q´1q br .) Now for u ∈ H 0 pS, G m q, the composite in SHpSq coincides with Thpu s 2´s1 id 1 q, which is the identity by (2.1.1) (recall that s 2´s1 is even), and in particular does not depend on u ∈ H 0 pS, G m q. Since the left and right arrows in the above composite are isomorphisms, we deduce that the middle arrow does not depend on u ∈ H 0 pS, G m q, which shows as above that the isomorphism (3.3.1.b) is independent of the choice of the trivialisation α. Let us come back to the general case, where L Ñ S is a possibly nontrivial line bundle. Let p : L˝Ñ S be the projection, and consider the tautological trivialisation τ of the line bundle p˚L over L˝. Let us consider the isomorphism ϕ : Th L˝p Bq " Ý Ñ Th L˝p Cq in SHpSqrη´1s, where ‚ B " 1 '2 , C " L '2 , ϕ " Thpτ '2 q in case (i). ‚ B " L bs 1 , C " L bs 2 , ϕ " Thpid L bs 1 bα bs 2´s1 q in case (ii), Let p 1 , p 2 : L˝ˆS L˝Ñ L˝be the projections, and set q " p˝p 1 " p˝p 2 . For i ∈ t1, 2u, the isomorphism pi ϕ : Th L˝ˆSL˝p Bq Ñ Th L˝ˆS L˝p Cq in SHpSqrη´1s is induced by the trivialisation pi τ of the line bundle q˚L over L˝ˆS L˝, 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 SHpSqrη´1s as well as a unique morphism g into the commutative diagram in SHpSqrη´1s Th L˝ˆS L˝p Cq As p : Th L˝p Bq Ñ Th S pBq and p : Th L˝p Cq Ñ Th S pCq are epimorphisms in SHpSqrη´1s (see (2.2.7)), it follows that f and g are mutually inverse isomorphisms in SHpSqrη´1s. 4.1.1. Let us consider the linear embeddings i k : P k Ñ P k`1 given by the vanishing of the last coordinate. Denote by ι k : S Ñ P k the S-point given by the composite 4.1.2. Assume given a collection D " pd 1 , . . . , d r q ∈ Z r for some r ∈ N. We will denote by OpDq the vector bundle Opd 1 q '¨¨¨' Opd r q over P k , for each k ∈ N. When k " 0, we have a canonical isomorphism OpDq ≃ 1 'r over P 0 " S. This yields, for any k ∈ N, a canonical map in SptpSq (4.1.2.a) Σ 2r,r 1 S " Th S p1 'r q ≃ Th P 0 pOpDqq ι k Ý Ñ Th P k pOpDqq.
(ii) If k and d are even, then (4.1.2.a) induces an isomorphism Σ 2r,r 1 S ≃ Th P k pOpDqq in SHpSqrη´1s. (iii) If k is even and d is odd, then Th P k pOpDqq ≃ Σ 2pk`rq,k`r 1 S . (iv) If k is odd and d is even, then Th P k pOpDqq ≃ Σ 2pk`rq,k`r 1 S ' Σ 2r,r 1 S .
Proof. Let us first prove (i). Assume that k and d are odd. Consider a linear embedding P 1 Ñ P k . Its normal bundle is Op1q 'k´1 , and its open complement is a vector bundle over P k´2 . The corresponding zero-section P k´2 Ñ P k P 1 induces an isomorphism in SHpSq, and is the restriction of a linear embedding P k´2 Ñ P k . Thus (1.4) yields a distinguished triangle in SHpSq Th P k´2 pOpDqq Ñ Th P k pOpDqq Ñ Th P 1 pOpDq ' Op1q 'k´1 q Ñ Σ 1,0 Th P k´2 pOpDqq.
Using induction on the odd integer k, we are reduced to assuming that k " 1. Now by (3.3.1.ii) we have in SHpSqrη´1s Th P 1 pOpDqq ≃ Σ 2s,s pTh P 1 pOp´1qq^r´sq, where s is the number of indices i ∈ t1, . . . , ru such that d i is even. Since d is odd, so is r´s, and using (3.3.1.i) we deduce that Th P 1 pOpDqq ≃ Σ 2pr´1q,r´1 Th P 1 pOp´1qq in SHpSqrη´1s.
But Th P 1 pOp´1qq vanishes in SHpSqrη´1s, because of the distinguished triangle (see nd the definition of the map η (recall that Op´1q˝" A 2 t0u). We have proved (i). Let us come back to the situation when k and d are arbitrary. Consider a linear embedding P k´1 Ñ P k avoiding the S-point ι k : S Ñ P k (we write P´1 " ∅). It is a closed immersion defined by the vanishing of a regular section of Op1q. Its open complement is isomorphic to A k , and the morphism j k : S Ñ A k induced by ι k induces an isomorphism in SHpSq. The canonical trivialisation of OpDq over P 0 " S is the restriction along j k of a trivialisation of OpDq| A k (induced by the trivialisation of Op1q| A k corresponding to the regular section of Op1q mentioned above). It follows that the map Th P 0 pOpDqq Ñ Th A k pOpDqq induced by j k induces an isomorphism in SHpSq. Thus (1.4) yields a distinguished triangle in SHpSq Consider now a linear embedding s : S " P 0 Ñ P k avoiding i k´1 pP k´1 q. Its open complement U is a line bundle over P k´1 . The corresponding zero-section P k´1 Ñ U induces an isomorphism in SHpSq, and is the restriction of the linear embedding i k´1 : P k´1 Ñ P k . Since the vector bundle s˚OpDq and the normal bundle s˚Op1q 'k to s are both trivial, we have by (1.4) a distinguished triangle in SHpSq
Therefore (iii) follows from (i). Finally, assume that k is odd and d is even. It follows from (ii) that the composite Th P k´1 pOpDqq Ñ Th P k pOpDqq Ñ Th P k`1 pOpDqq is an isomorphism in SHpSqrη´1s, hence Th P k´1 pOpDqq Ñ Th P k pOpDqq admits a retraction, giving a splitting of the triangle (4.1.3.a). In view of (ii), this proves (iv).

4.1.4.
Corollary. If k ∈ N is even, the structural morphism P k Ñ S induces an isomorphism Σ ∞ P k " Ý Ñ 1 S in SHpSqrη´1s.
Proof. The structural morphism is retraction of ι k , so the corollary follows from (4.1.3.ii) applied with r " 0.
4.1.5. Proposition. Let E, V 1 , . . . , V n be vector bundles of constant rank over S, and d 1 , . . . , d n ∈ Z. Assume that rank E is even and that d 1 rank V 1`¨¨¨`dn rank V n is odd. Then Th PpEq ppOpd 1 q b q˚V 1 q '¨¨¨' pOpd n q b q˚V n qq " 0 ∈ SHpSqrη´1s, where q : PpEq Ñ 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 , . . . , V n are all trivial. Then the statement follows from (4.1.3.i).
4.1.6. Proposition. Let E, V be vector bundles over S. Assume that E has constant odd rank. Then Th PpEq pV q Ñ Th S pV q is an isomorphism in SHpSqrη´1s. In particular Proof. By (1.10) and (1.9), this may be verified Zariski-locally on S, so we may assume that E and V are both trivial. Then the statement follows after suspending (4. We deduce the following splitting principle: 4.2.5. Corollary. Let X ∈ Sm S , and E Ñ X be a vector bundle of constant odd rank. Then there exists a morphism f : Y Ñ X in Sm S whose image in SHpSqrη´1s admits a section, and a vector bundle F Ñ Y such that f˚E ≃ F ' 1.
Proof. Applying the functor SHpXqrη´1s Ñ SHpSqrη´1s of (1.9) we may assume that X " S. Let us denote by p : pE _ q˝Ñ S the projection. Then p˚E _ admits a nowhere vanishing section s. Its dual s _ : p˚E Ñ 1 is surjective. Letting Q " ker s _ , we have an exact sequence of vector bundles over pE _ q( Then we may find an affine bundle g : Y Ñ pE _ q˝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 _ Ñ Q _ , see e.g. [Rio10, p.243]). Then Σ ∞ g : Σ ∞ Y Ñ Σ ∞ pE _ q˝is an isomorphism in SHpSq, and Σ ∞ p : Σ ∞ pE _ q˝Ñ 1 S admits a section in SHpSqrη´1s by (4.2.3). So we may set f " p˝g. Assume that E has constant rank r. Then in the notation of (1.5) and (1.6), we have in SHpSqrη´1s Thpu id E q " xu r y : Th S pEq Ñ Th S pEq.

Proposition. Let L Ñ S be a line bundle, and V Ñ S a vector bundle of constant rank r. If s ∈ Z is such that rs is even, then there exists an isomorphism in SHpSqrη´1s
Th S pV q ≃ Th S pV b L bs q.
Proof. Upon replacing V with V b L bs , we may assume that s ě 0. When α : 1 " Ý Ñ L is a trivialisation of the line bundle L over S we have an isomorphism in SHpSq

.a)
Thpid V bα bs q : Th S pV q Ñ Th S pV b L bs q.
Any trivialisation of L is of the form uα for some u ∈ H 0 pS, G m q, and we have in SHpSqrη´1s, by (4.3.1) and (2.1.1) Thpid V bpuαq bs q " Thpid V bα bs q˝Thpu s id V q " Thpid V bα bs q˝xu rs y " Thpid V bα bs q.
It follows that the image of the isomorphism (4.3.2.a) in SHpSqrη´1s is independent of the choice of the trivialisation α, and we conclude as in the proof of (3.3.1). In the sequel we will refer to a system pV m , U m , f m q as above as a model for BG, and use the notation E m G, B m G. 5.1.2. In the situation of (5.1.1), since B 1 G is cofibrant and each B m G Ñ B m`1 G is a cofibration (for the model structure of [MV99]), it follows that the colimit BG is canonically weakly equivalent to the homotopy colimit of the motivic spaces B m G in SpcpSq (see e.g. [Hir03,Theorem 19.9.1]). 5.1.3. Let G be a linear algebraic group over S, and choose a model for BG. Since the map colim m E m G Ñ S is a weak-equivalence of motivic spaces [MV99, Proposition 4.2.3], we obtain a canonical morphism S Ñ BG in HpSq. We say that the model is pointed if we are given an S-point of E 1 G. This yields map S Ñ BG in SpcpSq, whose image in HpSq is the canonical morphism described just above. [MV99,p.133].) Let us fix an integer n ∈ N and describe an explicit model for BGL n . Fix an integer p ě n (we will typically take p " n). For s ∈ N, we denote by Grpn, sq the grassmannian of rank n subbundles U ⊂ 1 's over S (for us a subbundle is locally split, so 1 's {U is a vector bundle). For each m ∈ N t0u, consider the S-scheme V m,p parametrising the vector bundles maps 1 'n Ñ 1 'pm ; then V m,p Ñ 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 'n a subbundle of 1 'pm ). Then the natural left GL n -action on 1 'n induces a right GL n -action on U m,p , which is free, and the quotient U m,p { GL n can be identified with the grassmannian Grpn, pmq. The inclusion 1 'm ⊂ 1 'm`1 given by the vanishing of the last coordinate induces an inclusion

(See also
which yields a GL n -equivariant morphism f m,p : U m,p Ñ U m`1,p . Then the family pV m,p , U m,p , f m,p q is an admissible gadget with a nice GL n -action. Indeed the first condition of [MV99, Definition 4.2.1] is satisfied because U 1,p possesses an S-point, and the second condition is satisfied with j " 2i. The fact that the group GL n is special implies the validity of condition (3) of [MV99, Definition 4.2.4]. We thus obtained a model for 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 'n . 5.1.5. Let H ⊂ 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 non-immediate point is condition (3) of [MV99, Definition 4.2.4]. So let F be a smooth S-scheme with a free right Haction. Consider the quotient E " pFˆGq{H, where the right H-action on G is given by letting h ∈ H act via g Þ Ñ h´1g. Right multiplication in G induces a free right G-action on E. For any U ∈ Sm S with a right G-action we have isomorphisms pEˆUq{G ≃ ppFˆGq{HˆUq{G ≃ pFˆpGˆUq{Gq{H ≃ pFˆUq{H, which are functorial in U, and thus permit to identify the morphisms pEˆUq{G Ñ E{G and pFˆUq{H Ñ 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 BG, we obtain a model for BH, where E m H " E m G with the induced H-action. This yields morphisms where T 1 " pTˆUq{G followed by where T 2 " T {G 1 . Each morphism is an epimorphism in the Nisnevich topology by assumption, hence so is their composite. Under this choice of a model for BG, we have

5.2.2.
Lemma. If G, G 1 are linear algebraic groups over S, we have an isomorphism Proof. Since the product with a given motivic space commutes with homotopy colimits, we have isomorphisms in HpSq BpGˆG 1 q ≃ hocolim m pB m GˆB m G 1 q by (5.2.1.a) and (5.

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 BG (see (5.1.1)). Assume given a character of G, that is a morphism of algebraic groups χ : G Ñ G m . Considering the right G-action on A 1 given by letting g ∈ G act via λ Þ Ñ χpgq´1λ, we define for each m ∈ N t0u a line bundle over B m G: (5.3.1.a) C m pχq " pE m GˆA 1 q{G.
The assignment χ Þ Ñ C m pχq satisfies
Let now H ⊂ G be a closed subgroup, and consider the morphisms B m H Ñ B m G defined in (5.1.5). If χ| H denotes the restriction of the character χ to H, then If G 1 is a linear algebraic group over S, lettingχ : GˆG 1 Ñ G χ Ý Ñ G m be the induced character of GˆG 1 , we have (5.3.1.d) C m pχq " C m pχqˆB m G 1 .

Let
G be a linear algebraic group over S, and χ a surjective character of G.
Letting H " ker χ, we thus have an exact sequence of algebraic groups over S Let us fix a model for BG. As explained in (5.1.5), this yields a model for BH, and morphisms p m : B m H Ñ B m G for m ∈ N t0u. By (5.3.1), we also have a line bundle C m pχq over B m G, such that C m pχq˝" ppE m GˆA 1 q{Gq˝" pE m GˆG m q{G " ppE m Gq{HˆG m q{G m " pE m Gq{H " B m H.
In view of (1.3.a), this yields a cofiber sequence in Spc ‚ pSq, for each m ∈ N t0u More generally (as in (1.4)), if V Ñ B m G is a vector bundle, we have a cofiber sequence in Spc ‚ pSq, (5.3.2.c) Th BmH pV q Ñ Th BmG pV q Ñ Th BmG pC m pχq ' V q.

In the situation of (5.3.2), let us define
Th BG pCpχqq " colim m Th BmG pC m pχqq ∈ Spc ‚ pSq.

5.3.4.
In the situation of (5.3.2), assume that the model for BG 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 1 G, 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 Ñ BG and B 1 H Ñ BH respectively, we obtain maps i : S Ñ BG and j : G m Ñ BH in SpcpSq. Note that i is the map described in (5.1.3), and that the left-hand cartesian square in (5.3.4.a) shows that the map j : G m Ñ BH classifies the H-torsor χ : G Ñ G m . The right-hand cartesian square in (5.3.4.a) shows that the G m -torsor B 1 H Ñ B 1 G pulls back to the trivial torsor along i 1 , which yields a trivialisation of the line bundle i1C 1 pχq over S, and thus a morphism in Spc ‚ pSq We thus obtain a commutative diagram in Spc ‚ pSq, whose rows are cofiber sequences pBHq`/ / pBGq`/ / Th BG pCpχqq 5.3.5. In the situation of (5.3.2), using (2.2.5) for the line bundle C m pχq Ñ B m G, and applying the functor Spc ‚ pB m Gqrη´1s Ñ Spc ‚ pSqrη´1s of (1.9), we have homotopy cocartesian squares in Spc ‚ pSqrη´1s T^pB m Hq / / Th BmG pCpχ m qq T^pB m Gq`/ /ẘ hich 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 SHpSqrη´1s 6. Computations of classifying spaces 6.1. Diagonalisable groups. Using the embeddings P k ⊂ P k`1 of (4.1.1) for k ∈ N, we define, for n ∈ Z P ∞ " hocolim k P k ∈ SpcpSq and Th P ∞ pOpnqq " hocolim k Th P k pOpnqq ∈ Spc ‚ pSq, and as usual write Th P ∞ pOpnqq ∈ SptpSq instead of Σ ∞ Th P ∞ pOpnqq. We have a natural map ι ∞ : S " P 0 Ñ P ∞ in SpcpSq. For each n ∈ Z, the line bundle Opnq over P 0 admits a canonical trivialisation, so that (4.1.2.a) yields a canonical map in SptpSq (6.1.0.a) Σ 2,1 1 S Ñ Th P ∞ pOpnqq.
Proof. We apply (4.1.3) with D " pnq, and so OpDq " Opnq. We obtain that Σ 2,1 1 S Ñ Th P k pOpnqq is a weak equivalence in SptpSqrη´1s when n, k are even. Taking the homotopy colimit over k yields a weak equivalence Σ 2,1 1 S Ñ Th P ∞ pOpnqq in SptpSqrη´1s when n is even. This proves (iii). The other statements are deduced in a similar way from (4.1.3).
6.1.2. Consider the model for BG m described in (5.1.4) with p " n " 1, under the identification G m " GL 1 . Then B m G m " P m´1 , and thus BG m " P ∞ . Furthermore, the line bundle C m pid Gm q over B m G defined in (5.3.1.a) may be identified with the tautological bundle Op´1q over P m´1 . 6.1.3. Theorem. Let n ∈ N t0u. The following hold in SHpSqrη´1s: (i) The natural morphism 1 S Ñ Σ ∞ BG m is an isomorphism.
(ii) If n is odd, the natural morphism 1 S Ñ Σ ∞ Bµ n is an isomorphism.
(iii) If n is even, the morphism G m Ñ Bµ n classifying the µ n -torsor G m Ñ G m given by taking n-th powers induces an isomorphism Σ ∞ G m ≃ Σ ∞ Bµ n .
Proof. Let us consider the model for BG m described in (6.1.2), where B m G m " P m´1 and BG m " P ∞ . Then the first statement follows from (6.1.1.i).
Next consider the character n : G m Ñ G m given by taking n-th powers. Its kernel is µ n , and the line bundle C m pnq over B m G m (defined in (5.3.1.a)) corresponds to the line bundle Op´nq over P m´1 (this may be seen for instance by combining (6.1.2) with (5. 3.1.b)). So we are in the situation of (5.3.2) with G " G m , χ " n, H " µ n . Thus (5.3.3.a) yields a distinguished triangle in SHpSq If n is odd, then Th P ∞ pOp´nqq " 0 in SHpSqrη´1s by (6.1.1.ii), and the above distinguished triangle shows that the morphism Σ ∞ Bµ n Ñ Σ ∞ BG m is an isomorphism in SHpSqrη´1s. Thus the second statement follows from the first.
By definition (see [PW18,§3]), we have SL c n " ker ν n . We view SL n as a subgroup of SL c n via the mapping M Þ Ñ pM, 1q. 6.2.1. Proposition. For n ∈ N t0u the inclusion SL n ⊂ SL c n induces an isomorphism Σ ∞ BSL n " Ý Ñ Σ ∞ BSL c n in SHpSqrη´1s. Proof. The character δ n : SL c n Ñ G m ; pM, tq Þ Ñ t is surjective (recall that n ě 1), and satisfies SL n " ker δ n .
Set P n " GL nˆGm , and denote by q n : P n Ñ G m the second projection. Let us fix an arbitrary model for BGL n , but choose the model for BG m described in (6.1.2), so that B m G m " P m´1 . Recall from (5.2.1.a) that this yields a model for BP n , such that B m P n " pB m GL n qˆpB m G m q " pB m GL n qˆP m´1 .
Letting g n : P n Ñ GL n be the first projection, we have, as characters P n Ñ G m ν n " gnpdet n q¨qnpid Gm q´2, where det n : GL n Ñ G m denotes the determinant morphism. It follows from (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) C m pν n q ≃ C m pdet n q b C m pid Gm q b´2 ; C m pq n q ≃ 1 b C m pid Gm q.
Recall from (6.1.2) that the line bundle C m pid Gm q Ñ B m G m corresponds to Op´1q Ñ P m´1 . Applying (4.1.5) to the projective bundle B m P n Ñ B m GL n , for m even we have (6.2.1.a) Th BmPn pC m pν n q ' C m pq n qq " 0 " Th BmPn pC m pq n qq ∈ SHpSqrη´1s.
Since the character δ n is the restriction of q n : P n Ñ G m , it follows from (5.3.1.c) that the line bundle C m pδ n q over B m SL c n is the pullback of C m pq n q over B m P n . By (5.3.2.c), we have a distinguished triangle in SHpSq Th Bm SL c n pC m pδ n qq Ñ Th BmPn pC m pq n qq Ñ Th BmPn pC m pν n q ' C m pq n qq Ñ Σ 1,0 Th Bm SL c n pC m pδ n qq. so that, in view of (6.2.1.a) Th Bm SL c n pC m pδ n qq " 0 ∈ SHpSqrη´1s for m even. Now the distinguished triangle in SHpSq (see (5.3.2.b)) SL n implies that, for m even, the natural map induces an isomorphism The statement follows by taking the homotopy colimit.
6.2.2. Remark. Let A ∈ SHpSq be an η-periodic commutative ring spectrum, and consider the corresponding cohomology theory A˚,˚p´q (see (3.1.1)). Then by (6.2.1), we have a natural isomorphism A˚,˚pBSL c n q ≃ A˚,˚pBSL n q. If A is SL-oriented (see (3.2.5)) and S " Spec k with k a field of characteristic not two, Ananyevskiy computed in [Ana15, Theorem 10] that A˚,˚pBSL n q " # A˚,˚pSqrrp 1 , . . . , p r´1 , ess h if n " 2r with r ∈ N t0u A˚,˚pSqrrp 1 , . . . , p r ss h if n " 2r`1 with r ∈ N, where p i has degree p4i, 2iq and e has degree p2r, rq (here the notation Rrrx 1 , . . . , x m ss h refers to the homogeneous power series ring in m variables over the graded ring R, see [Ana15,Definition 27]). This computation remains valid (with exactly the same arguments) when S is an arbitrary noetherian scheme of finite dimension, under the assumption that 2 is invertible in S. Removing that last assumption seems to require a modification of the arguments of [Ana15], the problem being with [Ana15, Lemma 6] (which is used to prove [Ana15, Theorem 9]). 6.3. GL and SL. In this section, we compare the classifying spaces BGL 2r , BGL 2r`1 , BSL 2r , BSL 2r`1 .
6.3.1. Recall that under the model for BGL n described in (5.1.4) for p " n, the scheme B m GL n is identified with the grassmannian Grpn, nmq. The closed immersion Grpn, nmq Ñ Grpn`1, pn`1qmq mapping a subbundle E ⊂ p1 'm q 'n to E ' 1 ⊂ p1 'm q 'n ' 1 'm " p1 'm q 'n`1 , where the inclusion 1 ⊂ 1 'm is given by the vanishing of the last m´1 coordinates, induces a morphism f m : B m GL n Ñ B m GL n`1 which is compatible with the transition maps as m varies. This yields a morphism in SpcpSq (6.3.1.a) BGL n Ñ BGL n`1 .
6.3.2. For integers u, v, w ∈ N, we denote by Grpu ⊂ v, wq the flag variety of subbundles P ⊂ Q ⊂ 1 'w with rank P " u and rank Q " v. Let r, s ∈ N, and consider the morphisms Grp2r, sq p Ð Ý Grp2r ⊂ 2r`1, sq q Ý Ñ Grp2r`1, sq given by mapping a flag P ⊂ Q to P , resp. Q.
For n ∈ t2r, 2r`1u, let us denote by U n ⊂ 1 's the tautological rank n subbundle over Grpn, sq, and write Q n " 1 's {U n . Then the morphism p may be identified with the projective bundle PpQ 2r q, and the morphism q is the projective bundle PpU _ 2r`1 q. 6.3.3. Proposition. The map BGL 2r Ñ BGL 2r`1 of (6.3.1.a) becomes an isomorphism in SHpSqrη´1s.
Proof. Let n " 2r. For m ∈ N t0u, consider the commutative diagram in Sm S Grpn, nmq Grpn ⊂ n`1, pn`1qmq where the morphism j m is given by mapping E ⊂ p1 'm q 'n to with the inclusion 1 ⊂ 1 '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 " pn`1qm. The morphism f m is the one described in (6.3.1), and the morphism g m is induced by the inclusion The morphisms of this diagram are compatible with the transition maps as m varies, induced by the inclusions 1 'm ⊂ 1 'm`1 given by the vanishing of the last coordinate. The morphism q m is a P n -bundle, hence is an isomorphism in SHpSqrη´1s by (4.1.6) (recall that n " 2r is even). The morphism p m is a P pn`1qm´n´1 -bundle, hence is also an isomorphism in SHpSqrη´1s when m is odd by (4.1.6).
In the notation of (5.1.4) the morphism g m is the GL n -quotient of the morphism U m,n Ñ U m,n`1 induced by (6.3.3.a). Therefore it follows from (5.1.1) that the map colim m g m is a weak equivalence of motivic spaces.
Applying the functor Σ ∞ : Sm S Ñ SptpSqrη´1s to the above diagram, and taking the homotopy colimit over m, we thus obtain a commutative diagram in SptpSqrη´1s, where all maps are weak equivalences. Since the map (6.3.1.a) is obtained as colim m f m , the proposition follows.
6.3.5. Using the model for BGL n described in (5.1.4) with p " n, the variety B m GL n coincides with the grassmannian Grpn, nmq. Observe that the tautological bundle U n over this variety is isomorphic to the quotient pE m GL nˆA n q{ GL n , where the right GL n -action on A n is given by letting ϕ ∈ GL n act via v Þ Ñ ϕ´1pvq. The GL n -equivariant isomorphism detpE m GL nˆA n q ≃ E m GL nˆA 1 , where the right GL n -action on A 1 is given by letting ϕ ∈ GL n act via λ Þ Ñ det n pϕ´1qλ, yields an isomorphism of line bundles over B m GL n (6.3.5.a) det U n ≃ C m pdet n q.
(Note that this permits to remove some of the technical assumptions present in the statement of [Lev19, Theorem 4.1].)
Thus, when n ą 0 is even, there is essentially one nontrivial invariant of µ n -torsors over S in SHpSqrη´1s, in the form of an element of End SHpSqrη´1s p1 S q. Moreover, it also follows from (6.1.3) that the morphism Bµ 2 Ñ Bµ 2r becomes an isomorphism in SHpSqrη´1s for r ą 0. Therefore the above-mentioned invariant of µ n -torsors is induced by an invariant of µ 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.3. Definition. It will be convenient to think of a µ 2 -torsor over S as a pair pL, λq, where L Ñ S is a line bundle, and λ : L " Ý Ñ L _ is an isomorphism of line bundles over S. Isomorphisms pL, λq Ñ pL 1 , λ 1 q are given by isomorphisms of line bundles ϕ : L " Ý Ñ L 1 such that λ " ϕ _˝λ1˝ϕ . The set of isomorphism classes of µ 2 -torsors is denoted H 1 et pS, µ 2 q; it is endowed with a group structure induced by the tensor product of line bundles.
A.4. Definition. Consider a µ 2 -torsor, given by a line bundle L Ñ S and an isomorphism λ : L " Ý Ñ L _ . Let us consider the composite isomorphism in SHpSq (see (A.1) for the definition of σ L , and (1.5) for that of Thpλq) This yields an element a pL,λq " Σ´Lpσ´1 L˝T hpλqq ∈ Aut SHpSq p1 S q. We define αpL, λq " pa p1,canq q´1˝a pL,λq ∈ Aut SHpSq p1 S q, where can : 1 Ñ 1 _ is the canonical isomorphism of line bundles over S. (The element a p1,canq corresponds to the element ǫ of [Mor04, §6.1].) A.5. This construction is compatible with pullbacks, in the sense that if f : R Ñ S is a morphism of noetherian schemes of finite dimension, and pL, λq a µ 2 -torsor, then the composite in SHpRq 1 R " f˚1 S f˚αpL,λq ÝÝÝÝÝ Ñ f˚1 S " 1 R is αpf˚L, f˚λq.
A.9. Remark. Let ρ : G m Ñ Bµ 2 be the map classifying the µ 2 -torsor p1, t idq over G m , where t ∈ H 0 pG m , G m q is the tautological section. Recall from (6.1.3.iii) that Σ ∞ ρ : Σ ∞ G m Ñ Σ ∞ Bµ 2 is an isomorphism in SHpSqrη´1s. If a µ 2 -torsor pL, λq is classified by the map f : S Ñ Bµ 2 , we claim that αpL, λq is the composite in SHpSqrη´1s (the map π 1 was defined in (2.2.1)) (A.9.a) 1 S Indeed, applying the functor SHpSqrη´1s Ñ SHpL˝qrη´1s which is faithful by (2.2.8), and using (A.7), we may assume that L " 1. Then λ is given by multiplication by an element u ∈ H 0 pS, G m q, and the map f factors as S u Ý Ñ G m ρ Ý Ñ Bµ 2 . Denoting by p : G m Ñ S is the projection, the composite (A.9.a) is given by which coincides with xuy ∈ Aut SHpSqrη´1s pSq. Thus the claim follows from (A.6).