Motivic cohomology of the Nisnevich classifying space of even Clifford groups

In this paper, we consider the split even Clifford group $\Gamma^+_n$ and compute the mod 2 motivic cohomology ring of its Nisnevich classifying space. The description we obtain is quite similar to the one provided for spin groups in [11]. The fundamental difference resides in the behaviour of the second subtle Stiefel-Whitney class that is non-trivial for even Clifford groups, while it vanished in the spin-case.


Introduction
Subtle characteristic classes were introduced by Smirnov and Vishik in [8] to approach the classification of quadratic forms by using motivic homotopical techniques.In particular, these characteristic classes arise as elements of the motivic cohomology ring of the Nisnevich classifying space BG of a linear algebraic group G over a field k.They naturally provide invariants for Nisnevich locally trivial G-torsors, which take value in the motivic cohomology of the base.What is probably more interesting is that they also provide invariants for étale locally trivial G-torsors, which take value this time in a more complicated and informative object, namely the motivic cohomology of the Čech simplicial scheme of the torsor under study.
In [8], the authors compute the motivic cohomology ring with Z /2-coefficients of BO n , i.e. the Nisnevich classifying space of the split orthogonal group.Similarly to the topological picture, this cohomology ring is a polynomial algebra over the motivic cohomology of the ground field generated by certain classes u 1 , . . ., u n called subtle Stiefel-Whitney classes.These invariants detect the power I n of the fundamental ideal of the Witt ring a quadratic form belongs to.In particular, the triviality of all subtle Stiefel-Whitney classes implies the triviality of the quadratic form itself. Besides, from the computation of H(BO n ) it follows that the mod 2 motivic cohomology of BSO n is also a polynomial algebra generated by all subtle Stiefel-Whitney classes but the first.
Following [8], we studied the motivic cohomology rings of the Nisnevich classifying spaces of unitary groups in [10], of spin groups in [11] and of projective general linear groups in [9].This paper is a natural follow-up of [11].In fact, we focus here on computing the motivic cohomology of the Nisnevich classifying space of even Clifford groups.These algebraic groups are closely related to spin groups.On the level of torsors, this is visible from the fact that a spin-torsor yields a quadratic form in I 3 through a surjective map with trivial kernel, while the torsors of the even Clifford groups are exactly the quadratic forms in I 3 .
The topological counterpart of the even Clifford group Γ + n is the Lie group Spin c (n).The singular cohomology of the classifying space of Spin c (n) was computed by Harada and Kono in [3].The main result we obtain in this article is a motivic version of [3,Theorem 3.5].More precisely, we prove the following.
Theorem 1.1.Let k be a field of characteristic different from 2 containing a square root of −1.Then, for any n ≥ 2, there exists a cohomology class e 2 l(n) in bidegree (2 l(n)−1 ) [2 l(n) ] such that the natural homomorphism of H-algebras is an isomorphism, where The assumption on the characteristic of the ground field is necessary since the mod 2 motivic cohomology of the point and the mod 2 motivic Steenrod algebra are well understood in characteristic different from 2 (see [14]).Moreover, we require also that k contains a square root of −1, since in this case the action of the mod 2 motivic Steenrod algebra on the mod 2 motivic cohomology of the point is trivial, making our computation easier.Anyways, we suspect that a result similar to Theorem 1.1 would still hold after dropping this last assumption, but with more complicated relations involving Steenrod operations and ρ = Sq 1 τ , where ρ is the class of −1 in mod 2 Milnor K-theory and τ is the generator of The similarity between Theorem 1.1 and the computation for the spin-case (see Theorem 2.10) is clear.Nonetheless, a crucial difference is that, while u 2 is trivial in H(BSpin n ), it is not in H(BΓ + n ) where the ideal of relations I • l(n) is generated by the action of the motivic Steenrod algebra over u 3 = Sq 1 u 2 .This also explains the gap between k(n) in Theorem 2.10 and l(n) in Theorem 1.1, which is due to discrepancies in the maximal length of the regular sequences in H(BSO n ) obtained by applying certain Steenrod operations to u 2 and u 3 respectively.
We conclude by pointing out that understanding the motivic cohomology of Nisnevich classifying spaces also helps in obtaining information about the structure of the Chow ring of étale classifying spaces B ét G (see [12]), which is an interesting object of study that is particularly challenging to fully grasp.For example, the Chow ring of B ét Γ + n has been recently investigated by Karpenko in [4] where he proves a conjecture that allows him, as a consequence, to compute the exponent indexes of spin grassmannians.In our case, we will show how to apply Theorem 1.1 to compute the subring generated by Chern classes of the Chow ring mod 2 of B ét Γ + n , modulo nilpotents, sheding new light on its complicated structure.
Outline.In Section 2, we report notations and preliminary results that we will use in this paper.In particular, we recall the Thom isomorphism in the triangulated category of motives over a simplicial base, which provides Gysin long exact sequences in motivic cohomology.Besides, we recall definitions and properties of classifying spaces in motivic homotopy theory, as well as the computation of the mod 2 motivic cohomology of BO n , BSO n and BSpin n .In Section 3, we investigate regular sequences in H(BSO n ) constructed starting from u 3 by acting with specific Steenrod operations.Finally, in Section 4, we exploit the Gysin sequence relating BΓ + n and BSpin n , and the regular sequences studied before, in order to fully compute the mod 2 motivic cohomology of BΓ + n .After that, we also obtain a complete description of the reduced Chern subring of the Chow ring mod where τ is the non-trivial class in H 0,1 ∼ = Z /2 and H n,n ∼ = K M n (k)/2.Note that, since we are working over a field containing the square root of −1, all Steenrod squares Sq i , as defined in [15], act trivially on H.
Since we will mainly work in the triangulated category of motives over a simplicial scheme defined by Voevodsky in [13], we recall a few definitions and propositions about it that will be useful later on to prove our main results.
Denote by CC(Y • ) the simplicial set obtained from Y • by applying the functor CC that sends any connected scheme to the point and respects coproducts.
a smooth coherent morphism of smooth simplicial schemes over k and A a smooth k-scheme such that: 1) X 0 is isomorphic to Y 0 × A and, under this isomorphism, π 0 becomes equal to the projection map The following result guarantees that the Thom isomorphism from Proposition 2.3 is functorial.
Proposition 2.5.Suppose there is a cartesian square such that Y 0 is connected, p X and p Y are smooth, π and π ′ are smooth coherent with fiber A satisfying all conditions from Proposition 2.3.Then, the induced homomorphism in motivic cohomology We now recall from [6] the definitions of the Nisnevich and étale classifying spaces of linear algebraic groups.
Let G be a linear algebraic group over k and EG the simplicial scheme defined by (EG) n = G n+1 , with partial projections as face maps and partial diagonals as degeneracy maps.The space EG is endowed with a right free G-action provided by the operation in G.
Definition 2.6.The Nisnevich classifying space of G is the quotient BG = EG/G.
The morphism of sites π : (Sm/k) ét → (Sm/k) N is induces an adjunction between simplicial homotopy categories Let H be an algebraic subgroup of G.Then, we can define the simplicial scheme BH = EG/H with respect to the embedding H ֒→ G. Denote by j the induced morphism BH → BH.Proposition 2.8.Let H ֒→ G be such that all rationally trivial H-torsors and G-torsors are Zariskilocally trivial.If the map Hom Hs(k) (Spec(K), B ét H) → Hom Hs(k) (Spec(K), B ét G) has trivial kernel for any finitely generated field extension K of k, then j is an isomorphism in H s (k).Remark 2.9.Note that the obvious map π : BH → BG is smooth coherent with fiber G/H.By using the Gysin sequence induced by the Thom isomorphism, one can compute by induction the following motivic cohomology rings.
Theorem 2.10.The motivic cohomology rings of BO n , BSO n and BSpin n are respectively given by where the ith subtle Stiefel-Whitney class u i is in bidegree

Regular sequences in H(BSO n )
In this section, we want to use the techniques developed in [11,Section 7] to produce other regular sequences in the motivic cohomology of BSO n that will be relevant later to deal with the case of even Clifford groups.
Let V be an n-dimensional Z /2-vector space, B a bilinear form over V and ⊥ V its right radical, i.e.
Fix a basis {e 1 , . . ., e n } for V and let x i and y j be the coordinates of x and y in V , respectively.Then, B(x, y) = n i,j=1 B(e i , e j )x i y j is a homogeneous polynomial of degree 2 in Z /2[x 1 , . . ., x n , y 1 , . . ., y n ].
Recall from [11, Section 7] that there are commutative squares x i is in bidegree (0) [1] and y i is in bidegree (1) [2] for any i, β n is obtained from α n by tensoring with Z /2 over H, γ n and δ n are the reduction homomorphisms along H → Z /2.
] be a ring homomorphism, where deg(b i ) = 1 for any i and f (a j ) is a homogeneous polynomial in B of positive degree α j for any j.Moreover, let r 1 , . . ., r k be a sequence of elements of A. If f (r 1 ), . . ., f (r k ) is a regular sequence in B, then r 1 , . . ., r k is a regular sequence in A.
Proof.Since θ j is inductively computed from θ 1 = u 3 by using only Wu formula (see [11,Proposition 5.7]) and Cartan formula, we know that θ j is an element of Z /2[τ, u 2 , . . ., u n ] for any j.The regularity of the sequence in Z /2[τ, u 2 , . . ., u n ] follows from Theorem 3.3 by noticing that, modulo τ and u 1 , . This clearly implies also the regularity of the sequence in H(BSO n ).
Recall from [11,Section 7] the homomorphisms i : , where i is the ring homomorphism defined by i(w i ) = u i , h is the linear map defined by h(x) = τ [ p i(x) 2 −q i(x) ] i(x) for any monomial x, where (q i(x) )[p i(x) ] is the bidegree of i(x), and t is the ring homomorphism defined by t(u i ) = w i , t(τ ) = 1 and t(K M r (k)/2) = 0 for any r > 0.
Lemma 3.5.For any homogeneous polynomials x and y in H top (BSO n ), we have that h(xy) = τ ǫ h(x)h(y), where ǫ is 1 if p i(x) p i(y) is odd and 0 otherwise.
] and e(∆ n ) is the Euler class of the complex spin representation ∆ n .
The following is the main result of this section.
By Theorem 3.8, we know that φ i ρ i for some homogeneous φ i ∈ H top (BSO n ) and, after applying h, we h(φ i )θ i by Lemmas 3.5 and 3.6.Thus, θ l(n) ∈ I • l(n) , which completes the proof.
Remark 3.10.Note that either l(n 4 The motivic cohomology ring of BΓ + n In this last section, we prove our main result that describes the structure of the motivic cohomology of the Nisnevich classifying space of even Clifford groups.Before proceeding, recall from [1, Section 3] that Γ + n -torsors are in one-to-one correspondence with quadratic forms with trivial discriminant and Clifford invariant, i.e. quadratic forms in I 3 , where I is the fundamental ideal of the Witt ring.Moreover, for any n ≥ 2, we have the following short exact sequences of algebraic groups (see [5, Chapter VI, Section 23.A]) Lemma 4.1.For any n ≥ 2, BSpin n ∼ = BSpin n with respect to the embedding Spin n ֒→ Γ + n .
Proof.First, note that, by [2] and [7], rationally trivial Spin n -torsors and Γ + n -torsors are locally trivial.Moreover, recall from [6, Section 4.1] that Hom Hs(k) (Spec(K), B ét G) ∼ = H 1 ét (K, G) for any Nisnevich sheaf of groups G. Therefore, it follows from [1, Section 3] that is surjective with trivial kernel, for any finitely generated field extension K of k.Hence, we can apply Proposition 2.8 to the case that G and H are respectively Γ + n and Spin n , which provides the aimed result.
Proposition 4.2.For any n ≥ 2, there exists a Gysin long exact sequence of H(BΓ + n )-modules From the motivic Serre spectral sequence (see [9,Theorem 5.12]) associated to the sequence it follows that H 1, * ′ (N ) ∼ = 0. Therefore, the homomorphism . By Proposition 2.3, Remark 2.9 and Lemma 4.1, it induces a Gysin long exact sequence of H(BΓ + n )modules We proceed by induction on p.For p = 0, the Serre spectral sequence associated to 2 implies that H 0, * ′ (BSO n ) ∼ = H 0, * ′ (BΓ + n ), which provides the induction basis.Now, suppose that H(BSO n ) → H(BΓ + n ) is surjective in topological degrees less than p < 2 k(n) , and consider a class x in H p, * ′ (BΓ + n ).Since by Theorem 2.10 the homomorphism Hence, by Proposition 4.2, there exists a class z in that is what we wanted to show.
Proof.This follows by noticing that x j maps to the respective class defined for spin groups in [11 2) If moreover ker(h * ) = im(g * p), then there is an isomorphism Proof.We start by proving 1).It immediately follows from Remark 4.5 and Lemma 4.7 that J • k(n) + (u 2 ω n ) ⊆ ker(p).We show the opposite inclusion by induction on the topological degree.Proposition 4.2 provides the induction basis.Now, suppose that x is in ker(p) and every class in ker(p) with topological degree less than the topological degree of x belongs to J • k(n) + (u 2 ω n ).We can write x as m j=0 φ j e j for some φ j ∈ H(BSO n ).Then, m j=0 φ j c j = g * p(x) = 0, and so φ j = 0 in H(BSpin n ) for any j since by hypothesis c is a monic polynomial in v 2 k(n) .Therefore, ) by Theorem 2.10.Hence, there are where z = m j=0 ψ j e j .Thus, u 2 p(z) = 0 which implies that p(z) ∈ im(h * ) = im(p) • ω n , and so there exists an element y in H We now move to 2).We prove by induction on the topological degree that, if ker(h * ) = im(g * p), then im(p) = H(BΓ + n ).Lemma 4.3 provides the induction basis.Let x be a class in H(BΓ + n ) and suppose that p is an epimorphism in topological degrees less than the topological degree of x.From g * (x) ∈ ker(h * ) = im(g * p) it follows that there is an element χ in H(BSO n ) ⊗ H H[e] such that g * (x) = g * p(χ).Therefore, x + p(χ) = u 2 z for some z ∈ H(BΓ Proof.We proceed by induction on m.For m = 0 we have that Sq 0 ω n = ω n and for m > 2 k(n) − 1 we have that Sq m ω n = 0 by [11,Corollary 5.8].Suppose that Sq i ω n ∈ ω n for i < m ≤ 2 k(n) − 1.Then, by Cartan formula, in H(BSO n ) we have that Sq m (u 2 ω n ) = u 2 Sq m ω n + τ u 3 Sq m−1 ω n + u 2 2 Sq m−2 ω n .

Definition 2 . 1 .
For any smooth simplicial scheme Y • over k, denote by c : Y • → Spec(k) the projection to the base.Then, we can define the Tate objects T

Proof. See [ 11 ,Definition 2 . 4 .
Proposition 4.2].We call Thom class of π and denote by α the image of 1 under the Thom isomorphism of Proposition 2.3.
is the ideal generated by θ 0 , . . ., θ k(n)−1 and k(n) depends on n as in the following table.

Definition 4 . 4 .
Denote by ω n the class h *

Remark 4 . 14 .
It immediately follows from Lemma 4.13 that h * (v j 2 k(n) 2 of B ét Γ + n .Sq 2 j−2 . . .Sq 2 Sq 1 w 2 in H top (BSO n ) θ j the element Sq 2 j−1 Sq 2 j−2 . . .Sq 2 Sq 1 u 2 in H(BSO n ) Γ + 5. It follows from Lemma 4.3 that ω n belongs to the image of H(BSO n ) → H(BΓ + n ).Moreover, Proposition 4.2 implies that u 2 ω n = 0 in H(BΓ + n ).Proposition 4.6.The motivic cohomology ring of BΓ + 2 is given by H(BΓ + 2 ) ∼ = H[u 2 , e 2 ], where e 2 is a lift of v 2 in H(BSpin 2 ) under the homomorphism H(BΓ + 2 ) → H(BSpin 2 ).Proof.Consider the Gysin long exact sequence from Proposition 4.2 Suppose there exists a class e in H(BΓ + n ) such that g * (e) is a monic homogeneous polynomial c in v 2 k(n) with coefficients in H(BSO n ), and denote by p the obvious homomorphism H + n ).By induction hypothesis z = p(ζ) for some element ζ ∈ H(BSO n ) ⊗ H H[e], hence x = p(χ + u 2 ζ) that is what we aimed to show.Remark 4.11.Since H(BSpin n ) is generated by the powers v i 2 k(n) as a H(BSO n )-module, we have that im(h * ) is generated by h * (v i 2 k(n) )as a H(BSO n )-module.Lemma 4.12.For any m ≥ 0, we have Sq m ω n ∈ ω n , where ω n is the H(BSO n )-submodule of H(BΓ + n ) generated by ω n .