Subtle characteristic classes for $Spin$-torsors

Extending [14], we obtain a complete description of the motivic cohomology with ${\mathbb Z}/2$-coefficients of the Nisnevich classifying space of the spin group $Spin_n$ associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel-Whitney classes in the motivic cohomology of \v{C}ech simplicial schemes associated to quadratic forms from $I^3$, which are closely related to $Spin_n$-torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel-Whitney class. Moreover, exploiting the relation between $Spin_7$ and $G_2$, we describe completely the motivic cohomology ring of the Nisnevich classifying space of $G_2$. The result in topology was obtained by Quillen in [13].


Introduction
Our main purpose in this work consists in an attempt of better understanding Spin-torsors, which are closely related to quadratic forms from I 3 . These are extremely interesting and fascinating objects and, although they arise quite naturally in many areas of mathematics, there are still many open questions about them due to their complexity and richness. In this paper, we try to study Spin-torsors from a motivic homotopic point of view by using classifying spaces and characteristic classes in motivic cohomology. At first, we need to mention that in the motivic homotopic environment there are two types of classifying spaces, the Nisnevich and theétale. The difference between the two is particularly visible when one works with non special algebraic groups. Indeed, in this case, the two types of classifying spaces above mentioned have in general different cohomology rings and, therefore, different characteristic classes. From [11], we know that torsors are classified byétale classifying spaces, nevertheless studying Nisnevich classifying spaces has shown to provide some advantages in the project of investigating them.
Actually, an essential inspiration for our work lies in [14], where the authors study torsors by using Nisnevich classifying spaces. They are mainly interested in BO n , the Nisnevich classifying space of the orthogonal group associated to the standard split quadratic form q n , which provides a key tool to study O n -torsors over the point which are nothing else but quadratic forms. In particular, they compute the motivic cohomology ring with Z /2-coefficients of BO n . This happens to be a polynomial algebra over the motivic cohomology of the point generated by some cohomology classes which are called subtle Stiefel-Whitney classes. These are very informative invariants, for example they enable to recognise the power of the fundamental ideal of the Witt ring where a quadratic form belongs and they are also connected to the J-invariant introduced in [17]. In a completely analogous way, it is possible to compute the motivic cohomology of BSO n , which again is a polynomial algebra generated by all the subtle Stiefel-Whitney classes but the first, as one would expect from the classical topological result.
In this work we go a bit further on this path by providing a complete description of the motivic cohomology with Z /2-coefficients of BSpin n , the Nisnevich classifying space of the spin group associated to the standard split form q n . As we have already mentioned, this is a step forward in the understanding of Spin-torsors, and so of quadratic forms with trivial discriminant and Clifford invariant. In topology the singular cohomology of BSpin n was computed by Quillen in [13]. Essentially, his computation is based on two key tools: 1) the regularity of a certain sequence in the cohomology ring of BSO n ; 2) the Serre spectral sequence associated to the fibration BSpin n → BSO n . Regarding 1), we essentially prove the regularity of a sequence in the motivic setting similar to Quillen's sequence in topology. This sequence is obtained from the second subtle Stiefel-Whitney class by acting with some specific Steenrod operations. As we will notice, the motivic situation is much more complicated than the topological one. This comes from the fact that in the motivic picture the element τ appears. Regarding 2), we use instead techniques developed in [14] to deal with fibrations of simplicial schemes with fibers which are motivically Tate, since in the motivic setting we lack a spectral sequence of Serre's type associated to a fibration. As a result, we get a description of the entire cohomology ring of BSpin n which is similar to the topological one in the same way as it is for the orthogonal and the special orthogonal cases. More precisely, we prove the following theorem (see Theorem 8.3). is an isomorphism, where I k(n) is the ideal generated by θ 0 , . . . , θ k(n)−1 and k(n) depends on n as in the table of Theorem 3.1.
Equivalently, one can visualize I k(n) as the ideal generated by the action of the motivic Steenrod algebra on the second subtle Stiefel-Whitney class. This way we obtain subtle classes for Spin-torsors and relations among them. Moreover, by exploiting the relation between Spin 7 and the exceptional group G 2 , we prove the following result that completely describes the motivic cohomology of BG 2 providing subtle characteristic classes for G 2 -torsors, namely octonion algebras (see Theorem 9.1). Since torsors are classified byétale classifying spaces, much attention has been devoted to investigate their Chow rings (see [16]), which neverthless are notoriously difficult to study. Regarding Spin n , the picture is completely understood for n ≤ 6 where the spin groups are known to be special by the sporadic isomorphisms. Guillot computed the Chow ring of the first non-trivial case, namely Spin 7 , together with the one of G 2 , over complex numbers in [4]. Next, Molina obtained the description of the Chow ring of Spin 8 over complex numbers in [9]. On the other hand, Yagita computed in [22] the whole motivic cohomology with Z /2-coefficients for Spin 7 and G 2 and provided a bound for the Chow ring with Z /2coefficients of all Spin n over complex numbers in [21] by exploiting Quillen's computation of the singular cohomology of BSpin n . In this paper we obtain a similar result by exploiting instead our computation of the motivic cohomology of the Nisnevich classifying space of Spin n which allows to relax the hypothesis on the base field and also suggests that understanding Nisnevich classifying spaces can possibly help in the study of theétale ones over more general fields.
Outline. We now shortly summarise the content of each section of this text. In Sections 2 and 3 we give some notations that we follow throughout this paper and recall some preliminary results from [13] regarding the computation of the cohomology ring of BSpin n in topology. In Section 4 we present some definitions and properties of the category of motives over a simplicial scheme which provide us with the main techniques essential to deal with fibrations of simplicial schemes with motivically Tate fibers. Section 5 is devoted to Nisnevich classifying spaces, to show some of their features and, in particular, to recall subtle Stiefel-Whitney classes. In Section 6 we construct a grid of long exact sequences involving the motivic cohomology of BSpin n and of BSO n which is our key tool, substituting the Serre spectral sequence, to get our main result. In Section 7 we show some results about regular sequences in H(BSO n ) obtained by acting with some Steenrod operations on the second subtle Stiefel-Whitney class, which allows us in Section 8 to prove the main theorem, i.e. the computation of the motivic cohomology ring of BSpin n . We see that, in general, this is not polynomial anymore in subtle Stiefel-Whitney classes, since many non trivial relations appear among them related to the action of the motivic Steenrod algebra on the second subtle Stiefel-Whitney class, and new subtle classes appear. Section 9 is devoted to the computation of the motivic cohomology ring of BG 2 . In Sections 10 and 11, using previous results, we find very simple relations among subtle classes in the motivic cohomology rings ofČech simplicial schemes associated to Spin-torsors and get some information about the Chern subring of the Chow ring with Z /2-coefficients of theétale classifying space of Spin n .
Acknowledgements. I would like to thank my PhD supervisor Alexander Vishik for his support and many helpful advice that made this text possible. I also want to thank the referee for very useful remarks that helped to improve the exposition.

Notation
Let us start in this section by fixing some notations we will use throughout this paper.
k field of characteristic not 2 containing √ −1 Spc(k), Spc * (k) category of motivic spaces over k, its pointed version H s (k), H s, * (k) simplicial homotopy category, its pointed version  , therefore Sq i (x) = 0 for i > 0 and for any x ∈ H n,n ∼ = K M n (k)/2, since H is trivial above the diagonal. Moreover, since we are working over a field containing the square root of −1, we have that Sq 1 τ = ρ = 0 where ρ is the class of −1 in K M (k)/2 and Sq i (τ ) = 0 for any i ≥ 2 by [20,Lemma 9.9]. It follows from this remark that, in our case, the only motivic cohomology operations that act non-trivially on H are the multiplications by elements of H.

Preliminary results
Our goal is to compute the motivic cohomology ring of the Nisnevich classifying space of Spin n , the spin group of the standard split quadratic form q n . In topology, the computation of the singular cohomology of BSpin n associated to the real euclidean quadratic form was achieved by Quillen in [13].
Before recalling his main results, let us define the elements ρ j in H top (BSO n ) ∼ = Z /2[w 2 , . . ., w n ] inductively by the following formulas: Moreover, we recall that the values written in the previous table are related to the dimension of spin representations of Spin n . More precisely, for any n there is a spin representation ∆ n : Spin n → SO 2 k(n) that induces a map B∆ n : BSpin n → BSO 2 k(n) on classifying spaces which, in turn, induces a homomorphism in cohomology B∆ * n : H top (BSO 2 k(n) ) → H top (BSpin n ). We denote by w i (∆ n ) the cohomology class B∆ * n (w i ) in H top (BSpin n ).
Theorem 3.2. Let I top k(n) be the ideal in H top (BSO n ) generated by the regular sequence from Theorem 3.1. Then, the canonical homomorphism is an isomorphism.
Furthermore, we notice that Theorem 3.2 relies on the Serre spectral sequence for the fibration B Z /2 → BSpin n → BSO n . In the motivic setting we do not have such a tool, so we use instead techniques developed by Smirnov and Vishik in [14] which we recall in the following sections.

Motives over a simplicial base
The main purpose of this section is to recall some key definitions and results regarding the triangulated category of motives over a simplicial base, which is an essential tool for our computation. Before starting, we would like to mention that the contents of this section are essentially the same as Section 3 in [15]. Here, there is only a further attention in the construction of all cofiber sequences at the level of motivic spaces first, which is needed for the compatibility with Steenrod operations. Moreover, there is the definition of Thom class and Corollary 4.4, which were not present in [15].
Let us fix a smooth simplicial scheme Y • over k and a commutative ring with identity R. Following [18], we denote by Sm/Y • the category in which objects are given by pairs (U, j), with j a non-negative integer and U a smooth scheme over Y j , and in which morphisms from (U, j) to (V, i) are given by pairs (f, θ), with θ : [i] → [j] a simplicial map and f : U → V a morphism of schemes, such that the following diagram is commutative Moreover, as for spaces over the point, let us denote by Spc(Y • ) = ∆ op Shv N is (Sm/Y • ) the category of motivic spaces over Y • and by Spc * (Y • ) its pointed counterpart, consisting of simplicial Nisnevich sheaves over Sm/Y • .
For any morphism f : where Cone(f ) is defined by the following push-out diagram in Spc * (Y • ) In [18] there is a construction of the category of motives over Y • with R-coefficients. This category is denoted by DM − ef f (Y • , R). We notice that every cofiber sequence in Spc * (Y • ) induces a distinguished triangle in DM − ef f (Y • , R). Besides, attached to this category there is a sequence of restriction functors The image of a motive N ∈ DM − ef f (Y • , R) under r * i is simply denoted by N i . Furthermore, we have the following adjunction for any morphism p : In the case that p is smooth, together with the previous one, there is also the following adjunction In particular, for any smooth simplicial scheme Y • over k, we have a pair of adjoint functors where c : Y • → Spec(k) is the projection to the base. Then, following [18,Section 5], one can define Tate ). At this point, we recall some facts about coherence taken from [14]. By a smooth coherent morphism we mean a smooth morphism π : X • → Y • such that there is a cartesian diagram is closed under taking cones and arbitrary direct sums. On the other hand, we have that Lπ # maps coherent objects to coherent ones for any smooth coherent morphism π.
In the following results, CC(Y • ) indicates the simplicial set built up from a simplicial scheme Y • by applying the functor CC sending any connected scheme to the point and commuting with coproducts.
The following proposition permits us to deal with fibrations of simplicial schemes with motivically Tate fibers. Proposition 4.1. Let Y • be a simplicial scheme, R be a commutative ring with identity, and suppose that the first singular cohomology group H 1 (CC(Y • ), R × ) is trivial. Let r, s be non-negative integers, and let N ∈ DM − coh (Y • , R) be a motive such that Proof. See [14, Proposition 3.1.5].
We point out that, for R = Z /2, the cohomology group H 1 (CC(Y • ), R × ) is always trivial. The next result is the core technique inspired by [14] that enables to generate long exact sequences in motivic cohomology, similar to Gysin sequences for sphere bundles in topology, for fibrations with motivically Tate fibers.
Proposition 4.2. Let π : X • → Y • be a smooth coherent morphism of smooth simplicial schemes over k and A a smooth k-scheme such that: where Cone(π) is the cone of π in Spc * (Y • ). Moreover, we get a Thom isomorphism of H(Y • , R)-modules Proof. In Spc * (Y • ) we have a cofiber sequence [1] in the motivic category DM − ef f (Y • , R). Since π is smooth coherent and π 0 is the projection Y 0 × A → Y 0 by hypothesis, we have that it is the projection over any simplicial component, i.e. π i is the projection . Hence, the map π i induces the projection . Moreover, we point out that M (Cone(π)) is a coherent motive, since both M (X • π − → Y • ) and T are coherent objects and DM − coh (Y • , R) is closed under taking cones. Since we are also assuming by hypothesis that H 1 (CC(Y • ), R × ) ∼ = 0 we can apply Proposition 4.1 to M (Cone(π)). Therefore, we obtain that M (Cone(π)) ∼ = T (r) [s] in DM − ef f (Y • , R), and the proof is complete.
The image of 1 under the Thom isomorphism is called Thom class and it is denoted by α.
Later on, we will also need the following proposition about functoriality of the Thom isomorphism.
• be smooth coherent morphisms of smooth simplicial schemes over k with Y 0 connected and A a smooth k-scheme that satisfies all conditions from the previous proposition with respect to π ′ and such that the following diagram is cartesian with all morphisms smooth Then, the induced square of motives in the category Proof. We start by noticing that in Spc * (Y ′ • ) we can complete our commutative diagram to a morphism of cofiber sequences where the isomorphisms in the third column follow by Proposition 4.2. If we restrict our previous diagrams to the 0 simplicial component we obtain in , as we aimed to show.
In particular, from the previous proposition it immediately follows the next corollary about functoriality of Thom classes.
sends α ′ to α, where α ′ and α are the respective Thom classes.

The Nisnevich classifying space
Throughout this paper, we are mainly interested in Nisnevich classifying spaces of linear algebraic groups over Spec(k). In this section we recall some of their properties and relations withétale classifying spaces. The contents of this section are similar to Section 4 in [15]. The main difference resides on the fact that, in order to deal with the Spin-case, it is essential to weaken the hypothesis in Proposition 5.1 from "is injective" (see [15,Proposition 4.1]) to "has trivial kernel". Moreover, here we have added Corollary 5.2, Proposition 5.6 and Corollary 5.8, which were not present in [15]. Given a linear algebraic group G over k, let us call by EG the simplicial scheme defined on simplicial components by (EG) n = G n+1 with partial projections and partial diagonals as face and degeneracy maps respectively. The operation in G induces a natural action on EG. Then, the Nisnevich classifying space BG is obtained by taking the quotient respect to this action, i.e. BG = EG/G. Moreover, from the morphism of sites π : (Sm/k)é t → (Sm/k) N is we obtain the following adjunction where π * is the restriction to Nisnevich topology and π * isétale sheafification. Then, a definition of theétale classifying space of G is provided by Bé t G = Rπ * π * BG . Although this definition presentś etale classifying spaces as objects of H s ((Sm/k) N is , there exists a geometric construction for their A 1homotopy type (see [11]) obtained from a faithful representation ρ : G ֒→ GL(V ) by taking the quotient respect to the diagonal action of G on an open subscheme of an infinite-dimensional affine space Now, let H be an algebraic subgroup of G. Then, we can define two simplicial objects related to BH, namely a bisimplicial scheme BH = (EH × EG)/H and a simplicial scheme BH = EG/H. We highlight that the obvious morphism of simplicial schemes π : BH → BG is trivial over each simplicial component with G/H-fibers. At this point, let us call by φ : BH → BH and ψ : BH → BH the two natural projections. Notice that φ is always trivial over each simplicial component with contractible fiber EG, therefore an isomorphism in H s (k). The behaviour of ψ is somewhat different. Indeed, we need to impose a precise condition in order to make it an isomorphism.
Proof. We start by noticing that the restriction of ψ over any simplicial component is given by the morphism (EH × G n+1 )/H → G n+1 /H. The simplicial scheme (EH × G n+1 )/H is nothing but theČech simplicial schemeČ(G n+1 → G n+1 /H) associated to the H-torsor G n+1 → G n+1 /H which becomes split once extended to G. In order to check thať is a simplicial weak equivalence it is enough, by [11,Lemma 1.11], to evaluate on henselian local rings. Therefore, we need to look at the morphism of simplicial setš for any henselian local ring R over k. Now, the fiber of G n+1 → G n+1 /H over any point Spec(R) of G n+1 /H is given by a H-torsor P → Spec(R) whose extension to G is split, so split itself by hypothesis. In other words, this fiber is nothing but the split H-torsor H × Spec(R) → Spec(R). In this way we have found a splitting of is a weak equivalence of simplicial sets, for any henselian local ring R. This implies that ψ is an isomorphism in H s (k).
In practice, in the case we are interested in, it is enough to check the hypothesis of the previous proposition only for field extensions of k. The reason resides on the fact that rationally trivial quadratic forms are Zariski-locally trivial (see [12,Theorem 5.1]). Indeed, we have the following corollary to the previous proposition.
Corollary 5.2. Let H and G be such that all rationally trivial H-torsors and G-torsors are Zariskilocally trivial. If the map Hom Hs(k) (Spec(K), Bé t H) → Hom Hs(k) (Spec(K), Bé t G) has trivial kernel for any field extension K of k, then ψ is an isomorphism in H s (k). In particular, BH ∼ = BH in H s (k).
Proof. Let R be any Henselian local ring over k and K its field of fractions. Then, we have the following commutative diagram Saying that all rationally trivial H-torsors and G-torsors are Zariski-locally trivial implies that the two vertical maps in the previous diagram have trivial kernels. Moreover, by hypothesis, we have that the bottom horizontal map has trivial kernel too. Therefore, the top horizontal map has trivial kernel and the statement follows by Proposition 5.1.
The natural embedding of algebraic groups H ֒→ G induces two morphisms j : BH → BH and g : BH → BG. The following result tells us that, under the hypothesis of the previous proposition, j identifies BH and BH in H s (k). Proof. We already know that in this case the morphisms of bisimplicial schemes φ and ψ become weak equivalences once restricted to simplicial components. It follows that the morphisms they induce on the respective diagonal simplicial objects, namely φ : ∆( BH) → BH and ψ : ∆( BH) → BH, are weak equivalences. So, in order to get the result, it is enough to provide a simplicial homotopy F for any n and any 0 ≤ i ≤ n. The reason why we would like to work with BH → BG instead of BH → BG is that the first is a coherent morphism which is trivial over the 0 simplicial component with fiber G/H. So, provided that the reduced motive of G/H is Tate, we could apply to it Proposition 4.2. In a nutshell, this is how one can reconstruct the cohomology of the Nisnevich classifying space of an algebraic group inductively by considering some filtration of it.
We now move our attention to some particular examples which are of main interest for the purposes of this paper. First, we recall that O n -torsors are in one-to-one correspondence with quadratic forms, SO n -torsors are in one-to-one correspondence with quadratic forms with trivial discriminant and Spin ntorsors yield quadratic forms with trivial discriminant and Clifford invariant via a surjective map with trivial kernel for n ≥ 2. Hence, we can apply Propositions 5.1 and 5.3 to the case that G and H are respectively O n+1 and O n , or SO n+1 and SO n , or Spin n+1 and Spin n for n ≥ 2. Moreover, we have the following short exact sequences of algebraic groups is the affine quadric defined by the equation q n+1 = 1. Now we recall that by [14,Proposition 3.1.3]. Hence, we can apply Proposition 4.2 to the fibrations we are mostly interested in, namely BO n → BO n+1 , BSO n → BSO n+1 and BSpin n → BSpin n+1 . Indeed, by exploiting the arguments above mentioned the following theorem is obtained in [14]. The generators u i are called subtle Stiefel-Whitney classes. It is possible to get the same description for H(BSO n ) with the only difference given by the fact that u 1 = 0. Indeed, one has the following result.
Proposition 5.6. The motivic cohomology ring of BSO n is completely described by Proof. It is enough to apply Proposition 4.2 to the coherent morphism BSO n → BO n whose fiber is isomorphic to µ 2 . This way one gets a Gysin long exact sequence of H(BO n )-modules in motivic cohomology Now, note that k * is a ring homomorphism, hence it sends 1 to 1. Since H 0,0 (BSO n ) ∼ = Z /2, it follows that l * is the 0 homomorphism in bidegree (0)[0]. This implies that j * sends 1 to u 1 . From the fact that it is a homomorphism of H(BO n )-modules we deduce that j * is the multiplication by u 1 . Hence, it is a monomorphism in all bidegrees, from which it follows that l * is the 0 homomorphism in all bidegrees. Therefore, k * is an epimorphism and it kills all monomials divisible by u 1 , from which we deduce that Unfortunately, as we will see, while for orthogonal and special orthogonal groups Gysin sequences are enough to get the description of the motivic cohomology of their classifying spaces, for spin groups this is not true anymore. Indeed, we need to use also the fibrations BSpin n → BSO n and study their induced homomorphisms in cohomology. We achieve this in the following sections.
We will also use the action of the motivic Steenrod algebra on subtle classes which is given by the following Wu formula as in the classical case.
From the previous result we immediately deduce the following corollary which will be useful in the next sections.
Then, Sq m w = w 2 and Sq j w = 0 for any j > m.
Proof. If m is even, by [20, Lemmas 9.8 and 9.9], there is nothing to prove since w is on the slope 2 diagonal. Consider m odd, then w = τ r−1 2 xu h 1 · · ·u hr where x is a monomial in even subtle classes and u h i are odd subtle classes (notice that r must be odd by degree reason). Therefore, by Cartan formula, we have that hr = w 2 since the monomial τ r−1 2 xu h 1 · · ·u h r−1 is on the slope 2 diagonal. Moreover, Sq j w = 0 for j > m for the same reason. 6 The fibration BSpin n → BSO n We have already noticed that the special orthogonal case does not differ much from the orthogonal one, at least from the cohomological perspective, in the sense that their motivic cohomology rings are both polynomial over the cohomology of the point in subtle Stiefel-Whitney classes. This is not true anymore for spin groups. The main reason is that in this case there are much more complicated relations among subtle classes given by the action of the motivic Steenrod algebra on u 2 which make the cohomology rings not polynomial in subtle Stiefel-Whitney classes anymore (precisely for n > 9) and, moreover, new classes appear. For this reason, in order to get our main result, together with an inductive argument we need to consider the fibration BSpin n → BSO n . More precisely, in order to investigate the motivic cohomology of BSpin n , we need to consider for any n ≥ 2 the commutative square where π and π are smooth coherent morphisms, trivial over simplicial components, with fiber isomorphic to the affine quadric A q n+1 defined by the equation q n+1 = 1.
In Spc * (BSO n+1 ) we can complete the previous diagram to the following one (commutative up to a sign in the right bottom square) where each row and each column is a cofiber sequence The previous diagram induces, in turn, a commutative diagram of long exact sequences in motivic cohomology with Z /2-coefficients, where all the homomorphisms are compatible with Steenrod operations and respect the H(BSO n+1 )-module structure. This remark comes from the fact that the following diagram of categories is commutative up to a natural equivalence and both functors in the right bottom corner have adjoints from the right, so we have the action of Steenrod operations on the motivic cohomology of objects belonging to the image of Spc * (BSO n+1 ) in DM − ef f (BSO n+1 , Z /2) pulled from H A 1 , * (k). Since Spin-torsors yield quadratic forms from I 3 via a map with trivial kernel and for quadratic forms Witt cancellation holds, we are allowed to use Propositions 5.
where all the homomorphisms are compatible with Steenrod operations and respect the H(BSO n+1 )module structure. We recall that, by applying Proposition 4.2 to the smooth coherent morphism π : BSO n → BSO n+1 , which has fiber isomorphic to A q n+1 whose reduced motive is Tate, there is a Thom isomorphism which sends 1 to the Thom class α. By Theorem 5.5, modulo this isomorphism f * is just the multiplication by the subtle Stiefel-Whitney class u n+1 , since it is the only class of its bidegree vanishing in H(BO n ). Since Spin n+1 /Spin n ∼ = A q n+1 , Proposition 4.2 applies also to the smooth coherent morphism π : BSpin n → BSpin n+1 . Therefore, we have a Thom isomorphism and a Thom class α ∈ H n+1,[ n+1 2 ] (Cone( π)). We notice that, by Corollary 4.4, α is nothing but the restriction of α from H n+1,[ n+1 2 ] (Cone(π)) to H n+1,[ n+1 2 ] (Cone( π)). Hence, modulo the Thom isomorphism, which induces an isomorphism Note that, from Theorem 5.5, the morphism h * is always the 0 homomorphism, which means at the same time that g * is surjective and f * is injective. From these remarks we obtain the next proposition.

Some regular sequences in H(BSO n )
The main aim of this section is to prove a result in the motivic setting similar to Theorem 3.1. We construct a sequence θ 0 , θ 1 , . . . , θ k(n)−1 in H(BSO n ) by applying some Steenrod operations to u 2 just as in the topological case. Then, we focus on the two sequences obtained from the previous one by imposing on the one hand τ = 1 and on the other τ = 0. While the regularity of the first sequence was completely established by Quillen, nothing was known about the regularity of the second. We follow Quillen's method which allows to obtain the regularity of the sequence in topology by studying it in the cohomology of a certain power of BO 1 where it has an easier shape, related to some quadratic form over Z /2. The lenght k(n) of the regular sequence essentially depends on the characteristics of this quadratic form. For τ = 0, this approach does not work completely, so we study instead our sequence in the cohomology of a certain power of BO 2 . In this ring our sequence has a simple form, related now to a certain bilinear form over Z /2. As for the topological case, by studying these bilinear forms, we are able to get the regularity of some sequences of lenght h(n) (related to our initial motivic sequences) with τ = 0. Surprisingly, these sequences are either long as Quillen's sequences or have one less element. Then, combining Quillen's result (τ = 1) with ours (τ = 0), we get the regularity of θ 0 , θ 1 , . . . , θ k(n)−1 in the motivic cohomology of BSO n for the same values that appear in topology.
Let V be an n-dimensional Z /2-vector space and Ω an algebraically closed field extension of Z /2. We denote by V Ω the Ω-vector space Ω ⊗ Z /2 V . Note that the Frobenius automorphism acts on V Ω via the first tensor factor. Following [13], we also denote by x → x 2 the Frobenius transformation on V Ω . First, we recall the following result from [13]. Note that B(x, y) can be seen as a homogeneous element of degree 2 in Ω[x 1 , . . . , x n , y 1 , . . . , y n ]. In fact, let {e 1 , . . . , e n } be a basis for V , then B(x, y) = n i,j=1 B(e i , e j )x i y j where the x i and the y j are the coordinates of x and y respectively in the chosen basis. Let h = n − dim( ⊥ V ) and consider the ideal J in Ω[x 1 , . . . , x n , y 1 , . . . , y n ] generated by the homogeneous polynomials B(x, y), B(x, y 2 ), . . . , B(x, y 2 h−1 ).
Proposition 7.2. The algebraic subset in V Ω × V Ω defined by the ideal J is given by Proof. From Proposition 7.1 we know that W Ω is stable under the Frobenius transformation for any subspace W of V . Hence, if (x, y) belongs to W ⊥ Ω × W Ω , then y, y 2 , . . . , y 2 h−1 are in W Ω and x ∈ W ⊥ Ω . It follows that B(x, y), B(x, y 2 ), . . . , B(x, y 2 h−1 ) are all zero, so (x, y) ∈ V ar(J). Therefore, On the other hand, let (x, y) be a point of V ar(J) and consider the subspace M y of V Ω defined by M y = y, y 2 , . . . , Obviously, (x, y) belongs to M ⊥ y × M y . In order to prove that M y is of the form W Ω for some W ⊆ V it is enough to show that M y is stable under the Frobenius transformation. Note that ⊥ V Ω is stable under the Frobenius transformation, and so, if y 2 i ∈ ⊥ V Ω for some i, then y 2 j ∈ ⊥ V Ω for all j ≥ i. Hence, M y / ⊥ V Ω = y, y 2 , . . . , y 2 i−1 and y 2 i , . . . , y 2 h−1 ⊆ ⊥ V Ω for some i ≤ h. If i < h, then M y is stable under the Frobenius transformation, since ⊥ V Ω is so. If i = h, then M y = y, y 2 , . . . , y 2 h−1 ⊕ ⊥ V Ω . Therefore, if y, y 2 , . . . , y 2 h−1 are linearly independent then M y = V Ω since dim( ⊥ V ) = n − h, so y 2 h clearly belongs to M y . Otherwise, y 2 i ∈ y, . . . , y 2 i−1 for some 0 ≤ i ≤ h − 1, from which it follows that y 2 h ∈ y, . . . , y 2 h−1 . Hence, M y is stable under the Frobenius transformation from which we deduce that which completes the proof.
From the previous proposition we immediately obtain the following result.
Before proceeding we need the following technical lemma on regular sequences. j . Note that hi = id B , f = hg and g(a j ) is homogeneous in C for any j. The sequence b 1 , . . . , b n , g(a 1 ), . . . , g(a m ) is regular in C, so it is g(a 1 ), . . . , g(a m ) since regular sequences of homogeneous elements of positive degree permute (see for example [2,Corollary 17.2]). From [5,Proposition 1] it follows that C is a free A-module. At this point, note that c 1 , . . . , c m , if (r 1 ), . . . , if (r k ) is a regular sequence in C essentially by hypothesis. Hence, the sequence g(r 1 ), . . . , g(r k ) is regular in C, since g(r j ) + if (r j ) ∈ ker(h) = (c 1 , . . . , c m ) for any j. The fact that C is a free A-module via g implies that r 1 , . . . , r k is regular in A, which is what we aimed to show.

Corollary 7.3 implies that the sequence
Therefore, the sequence is regular in S 2m for the same values of h(2m). Now, consider the case n = 2m + 1. Similarly to the previous case, In this case, dim( ⊥ V ) = 1. In fact, from B(e i , y) = 0 it follows that y i = 0 for any i ≤ m − 1. Hence, ⊥ V = (0, . . . , 0, 1) , from which it follows by Corollary 7.3 that the sequence is a regular sequence in S 2m+1 , where h(2m + 1) = m. This completes the proof.
Define the elements θ j in H(BSO n ) inductively by the following formulas: Corollary 7.6. The sequence τ, θ 0 , . . . , θ h(n)−1 is regular in H(BSO n ), where h(n) depends on n as in the table of Theorem 7.5.
Proof. First, note that all θ j are obtained from u 2 by using only Wu formula (Proposition 5.7) and Cartan formula where elements of K M (k)/2 are never involved, from which it follows that every θ j is an element of Z /2[τ, u 2 , . . . , u n ]. Since K M (k)/2 is free over Z /2, it is enough to show the regularity of the sequence in Z /2[τ, u 2 , . . . , u n ]. Then, the result follows from Theorem 7.5 by noticing that, modulo τ and u 1 , θ j = i n γ n (Sq 2 j−1 · · · Sq 1 u 2 ), where i n is the inclusion of S n in H(BO n ).
is the bidegree of i(x), and extending linearly and t by imposing t(u i ) = w i , t(τ ) = 1 and t(K M r (k)/2) = 0 for any r > 0 and extending to a ring homomorphism.
We start by describing some properties of these homomorphisms. First of all, i and h are graded with respect to the usual grading in H top (BSO n ) and the topological degree in H(BSO n ). Besides, by the very definition of h, h(x) has bidegree ([ ] for any homogeneous polynomial x. On the other hand, we notice that h is not a ring homomorphism. Anyway, we have the following lemmas. Proof. At first consider two monomials x and y. Then, we get where ǫ jk is 1 if p i(x j ) p i(y k ) is odd and 0 otherwise. Now, we recall that p i(x j ) = p i(x) and p i(y k ) = p i(y) for any j and k, from which it immediately follows that h(xy) = τ ǫ l Notice that z j = τ n j x j , for some monomials x j in Z /2[u 2 , . . . , u n ]. By the very definition of i and t we get that it(z j ) = x j . Thus, since p x j = p z j = p z and q x j + n j = q z j = q z . Lemma 7.9. For any j, t(θ j ) = ρ j and h(ρ j ) = θ j .
Proof. Since a Wu formula (Proposition 5.7) holds even in the motivic case by 5.7, we get that t(θ j ) = ρ j by the very definition of t. Then, h(ρ j ) = ht(θ j ) = θ j by Lemma 7.8 and by recalling that θ j is in bidegree At this point, denote by I j the ideal in H(BSO n ) generated by θ 0 , . . . , θ j−1 and by I top j the ideal in H top (BSO n ) generated by ρ 0 , . . . , ρ j−1 . We are now ready to prove the main result of this section.
Proof. Since K M (k)/2 is free over Z /2 we just need to show the regularity of the needed sequence in Z /2[τ, u 2 , . . . , u n ]. From the fact that regular sequences of homogeneous elements of positive degree permute and by Corollary 7.6, we immediately deduce the regularity of the sequence for n = 0, 1, 2, 6 and 7(mod 8), since in these cases h(n) = k(n). Now, suppose n = 3, 4 or 5(mod 8). In these cases, h(n) = k(n) − 1, therefore Corollary 7.6 implies that the sequence τ, θ 0 , . . . , θ k(n)−2 is regular. Let z be a homogeneous polynomial in Z /2[τ, u 2 , . . . , u n ] such that zθ k(n)−1 ∈ I k(n)−1 . Then, we deduce that t(z)ρ k(n)−1 ∈ I top k(n)−1 . It follows from Theorem 3.1 that t(z) = k(n)−2 l=0 ψ l ρ l for some homogeneous ψ l ∈ H top (BSO n ) and, after applying h, we obtain τ [ pz 2 −qz] z = k(n)−2 l=0 τ ǫ l h(ψ l )θ l by Lemmas 7.7, 7.8 and 7.9. Hence, the regularity of θ 0 , . . . , θ k(n)−2 , τ implies that z ∈ I k(n)−1 and we obtain the regularity of the sequence θ 0 , . . . , θ k(n)−1 . At this point, we only need to show that θ k(n) ∈ I k(n) . Note that ρ k(n) ∈ I top k(n) by Theorem 3.2. Hence, ρ k(n) = k(n)−1 l=0 φ l ρ l for some homogeneous φ l ∈ H top (BSO n ) and, after applying h, we obtain θ k(n) = k(n)−1 l=0 τ ǫ l h(φ l )θ l by Lemmas 7.7 and 7.9. Thus, θ k(n) ∈ I k(n) , which completes the proof. 8 The motivic cohomology ring of BSpin n In this section we prove a motivic version of Theorem 3.2. The general strategy consists in using the grid of long exact sequences in motivic cohomology from Diagram 1 in Section 6 in order to get the result by an inductive argument. This method allows us to lift, not only subtle classes, but even relations among them from the cohomology of BSpin n to the cohomology of BSpin n+1 . These relations are essentially the elements θ j of the motivic regular sequences encountered in the previous section. Moreover, we see that a new subtle class v 2 k(n) appears in the motivic cohomology of BSpin n and the obstruction to lift it to the cohomology of BSpin n+1 is detected by the increasing of the lenght of the regular sequence moving from n to n + 1. In the proof of the main theorem it is essential to deal with the two possible cases separately: on the one hand the case that v 2 k(n) is liftable and the lenght of the regular sequence stays unchanged, i.e. k(n + 1) = k(n), on the other the case that v 2 k(n) is not liftable and the lenght of the regular sequence increases by one, i.e. k(n + 1) = k(n) + 1. Furthermore, we notice that when v 2 k(n) is not liftable, then "almost" its square is so, giving rise to a new extra class v 2 k(n)+1 in doubled degrees.
We start by showing that, as in topology, the second subtle Stiefel-Whitney class is trivial in the motivic cohomology ring H(BSpin n ).
Lemma 8.1. For any n ≥ 2, u 2 is trivial in H(BSpin n ). Moreover, there exists a unique element x 0 in H(Cone(a n )) such that b * n (x 0 ) = u 2 .
Proof. Recall that SO 2 ∼ = Spin 2 ∼ = G m , where G m is the multiplicative group, and the morphism from − − → G m , which induces the map on classifying spaces a 2 : BG m → BG m . By Kummer theory, the induced morphism on Picard groups P ic(BG m ) → P ic(BG m ) is multiplication by 2. Now, recall that P ic(BG m ) ∼ = H 2,1 (BG m , Z) (see [8,Corollary 4.2]). Then, for n = 2 the homomorphism a * 2 : Now, suppose u 2 = 0 in H(BSpin n ), then u 2 should be divisible by u n+1 in H(BSpin n+1 ), which forces u 2 to be trivial by degree reasons. Therefore, by induction, u 2 = 0 in H(BSpin n ) for any n. It immediately follows that there exists x 0 in H(Cone(a n )) such that b * n (x 0 ) = u 2 for any n ≥ 2. We prove its uniqueness by showing that b * n is a monomorphism in bidegree (1) [2]. First of all we notice that, for any n ≥ 2, H 1,1 (BSpin n ) ∼ = K M 1 (k)/2 by induction on n and by observing that g * is an isomorphism in bidegree (1) [1]. Hence, c * n : H 1,1 (BSpin n ) → H 2,1 (Cone(a n )) is the zero homomorphism, since the composition H 1,1 → H 1,1 (BSO n ) → H 1,1 (BSpin n ) is surjective and, therefore, so is the second map. It follows that b * n : H 2,1 (Cone(a n )) → H 2,1 (BSO n ) is a monomorphism, as we aimed to show.
From the previous lemma, for any n ≥ 2, we have a canonical set of elements x j in H(Cone(a n )) defined by x j = Sq 2 j−1 · · · Sq 1 x 0 for any j > 0. Denote by x 0 , . . . , x j−1 the H(BSO n )-submodule of H(Cone(a n )) generated by x 0 , . . . , x j−1 . Before proceeding we need the following lemma. Lemma 8.2. For any j ≥ 1, x j / ∈ x 0 , . . . , x j−1 in H(Cone(a 2 )), and consequently in any H(Cone(a n )).
Proof. We start by considering the Bockstein homomorphism β associated to the short exact sequence 0 → Z → Z → Z /2 → 0. The homomorphism a * 2 on cohomology with integer coefficients sends u 2 to 2v 2 where v 2 is the generator of H(BSpin 2 ) ∼ = H(BG m ) and so is injective, hence b * 2 is the 0 homomorphism on cohomology with integer coefficients, from which it follows that x 0 cannot come from any integral cohomology class. Thus, y = β(x 0 ) = 0. Moreover, since u 2 comes from an integral cohomology class, we have b * 2 (y) = 0, so y = mc * 2 (v 2 ) for some integer m. At this point we notice that mv 2 is in the image of a * 2 for any even m, so m must be odd, which implies that y is not divisible by 2, since v 2 mod (2) is not in the image of a * 2 . This is enough to conclude that Hence, x 1 = c * 2 (v 2 ) from which we deduce that by [20,Lemma 9.8], since Then, we would have that that is impossible since c * 2 is injective on the slope 2 line (above zero), which comes from the fact that H(BSO 2 ) ∼ = H[u 2 ] and a * 2 (u 2 ) = 0.
At this point, we are ready to prove our main result which provides the complete description of the motivic cohomology of BSpin n over fields of characteristic different from 2 containing √ −1.
Theorem 8.3. For any n ≥ 2, there exists a cohomology class v 2 k(n) of bidegree (2 k(n)−1 )[2 k(n) ] such that the natural homomorphism of H-algebras is an isomorphism, where I k(n) is the ideal generated by θ 0 , . . . , θ k(n)−1 and k(n) depends on n as in the table of Theorem 3.1.
Proof. Our proof goes by induction on n, starting from n = 2. Inductive step: We denote by θ ′ j and θ j the class Sq 2 j−1 · · · Sq 1 u 2 in H(BSO n ) and H(BSO n+1 ) respectively, by I ′ k(n) the ideal generated by the elements u 2 , θ ′ 1 , . . . , θ ′ k(n)−1 , by I k(n) the ideal generated by u 2 , θ 1 , . . . , θ k(n)−1 , by x ′ 0 and x 0 the unique lifts of u 2 to H(Cone(a n )) and H(Cone(a n+1 )) respectively, by x ′ j the class Sq 2 j−1 · · · Sq 1 x ′ 0 and by x j the class Sq 2 j−1 · · · Sq 1 x 0 . Now, suppose by induction hypothesis that we have an isomorphism where k(n) is the value prescribed by the table of Theorem 3.1.
Looking at the long exact sequence from Diagram 1 in Section 6 and by induction on degree we know that, in square degrees less than 2 k(n) , in H(BSpin n+1 ) there are only subtle Stiefel-Whitney classes, i.e. the homomorphism a * n+1 : , where α is the Thom class of the morphism g. We point out that The following result, whose proof is reported at the end of this section, enables to complete the induction step. It is indeed the core proposition that permits to conduct the proof of our main theorem.
If moreover ker( h * ) = Im( g * p n+1 ), then we get an isomorphism So, in order to finalize the proof we only need to find a cohomology class v which satisfies the requirements of Proposition 8.4. There are two possible cases: 1) h * (v 2 k(n) ) = 0; 2) h * (v 2 k(n) ) = 0.
Case 1 : In this case v 2 k(n) can be lifted to H(BSpin n+1 ) so w = 0 and we can choose c = v 2 k(n) . It follows that Im( h * ) = 0 = Im(p n+1 )· h * (v 2 k(n) ) and ker( h * ) = H(BSpin n ) = Im(p n ) = Im(p n (g * ⊗ l)) = Im( g * p n+1 ), since in this case p n and g * ⊗ l are surjective. So, by Proposition 8.4, we have that the homomorphism is an isomorphism. Furthermore, we observe that k(n + 1) = k(n) is the value predicted by the table of Theorem 3.1 since θ k(n) ∈ I k(n) as it is zero in H(BSpin n+1 ) (because u 2 is). This completes the first case. Case 2 : In this case we notice that the element w such that a * n+1 (w) α = h * (v 2 k(n) ) must be different from 0.
Remark 8.5. Since H(BSpin n ) is generated by v i 2 k(n) as a H(BSO n )-module (and, so, as a H(BSO n+1 )module) by induction hypothesis, we have that Im( h * ) is generated by h * (v i 2 k(n) ) as a H(BSO n+1 )module.
At this point, we need the following lemmas whose proofs are reported at the end of this section. Lemma 8.6. For any m, we have Sq m a * n+1 (w) ∈ a * n+1 (w) , where a * n+1 (w) is the H(BSO n+1 )submodule of H(BSpin n+1 ) generated by a * n+1 (w).
In order to prove the last part of the proposition we show by induction on degree that, if ker( h * ) = Im( g * p n+1 ), then p n+1 is surjective. The induction basis comes from the fact that, in square degree ≤ 2, H(BSpin n+1 ) is the same as the cohomology of the point. Take an element x and suppose that p n+1 is surjective in square degrees less than the square degree of x. From g * (x) ∈ ker( h * ) = Im( g * p n+1 ) it follows that there is an element χ in H(BSO n+1 ) ⊗ H H[v] such that g * (x) = g * p n+1 (χ). Therefore, x + p n+1 (χ) = u n+1 z for some z ∈ H(BSpin n+1 ). By induction hypothesis z = p n+1 (ζ) for some element , which is what we aimed to show.
Proof of Lemma 8.6. We proceed by induction on m. For m = 0 there is nothing to prove and for m > 2 k(n) − n we have that Sq m w = 0 by Corollary 5.8. Suppose the statement is true for integers less than m ≤ 2 k(n) − n. Then, from which it follows, by applying a * n+1 and by noting that u n+1 a * n+1 (w) = 0, that where all the elements but one in the sum disappear since by induction (on m) hypothesis Sq m−j a * n+1 (w) ∈ a * n+1 (w) for j > 0 and u n+1 a * n+1 (w) = 0. Hence, f * (Sq m a * n+1 (w) α) = 0, from which it follows that Sq m a * n+1 (w) α ∈ Im( h * ). By Remark 8.5, we obtain that Sq m a * n+1 (w) α = i≥1 φ i h * (v i 2 k(n) ) for some φ i ∈ H(BSO n+1 ). But, for any i > 1, the square degree of h * (v i 2 k(n) ) is greater than that of Sq m a * n+1 (w) α. We deduce that Sq m a * n+1 (w) α = φ 1 h * (v 2 k(n) ), from which it follows that Sq m a * n+1 (w) = φ 1 a * n+1 (w) ∈ a * n+1 (w) which is what we aimed to prove.
Note that there is a commutative diagram where  In this section we deduce, just from the triviality of u 2 in the motivic cohomology of BSpin n , some very simple relations among subtle classes in the motivic cohomology of theČech simplicial scheme associated to a Spin n -torsor. This provides information about the kernel invariant (see [14, 2.7.1]) of quadratic forms from I 3 .
We start by recalling that there exists a map from Spin n -torsors over the point to n-dimensional quadratic forms from I 3 which is surjective and has trivial kernel, where I is the fundamental ideal in the Witt ring. Moreover, we have the following commutative diagram for any n-dimensional q ∈ I 3 and all above-diagonal classes in H(BSpin n ) coming from theétale classifying space trivialise in H(Č(X q )), since the above-diagonal cohomology of a point is zero. HereČ(X q ) is theČech simplicial scheme associated to the torsor X q = Iso{q ↔ q n }. In particular Chern classes c i (q) = τ i mod2 u i (q) 2 are zero, as these are coming from theétale space (see [14] just before Thorem 3.1.1). From previous remarks we obtain the following proposition, which provides us with relations among subtle characteristic classes for quadratic forms from I 3 .
In [14], Smirnov and Vishik highlighted the deep relation between subtle Stiefel-Whitney classes and the J-invariant of quadrics defined in [17]. More precisely, they proved the following result.
From the previous theorem and from Proposition 10.1 we immediately deduce the following well known corollary. Corollary 10.3. For any n-dimensional q ∈ I 3 , 2 j−1 ∈ J(q) for any j satisfying 2 j + 1 ≤ n.
11 The Chern subring of Ch(Bé t Spin n ) In this last section we obtain from the structure of H(BSpin n ) some information about the subring generated by Chern classes (coming from the representation given by the map Spin n → SO n ) of the Chow ring Ch(Bé t Spin n ). This is a generalization to more general fields of a result by Yagita (see [ Lemma 11.1. The homomorphism H(Bé t SO n ) → H(Bé t Spin n ) maps w 2 to 0.
Note, however, that c 2 is not always mapped to 0 in H(Bé t Spin n ) as the computations of Ch(Bé t Spin 7 ) in [4] and of Ch(Bé t Spin 8 ) in [9] show. This implies that c 2 is non zero in Ch(Bé t Spin n ) for all n ≥ 7 just by looking at the homomorphisms Ch(Bé t Spin n ) → Ch(Bé t Spin n−1 ) that send c i to c i for all i ≤ n − 1.
In the following lemma we report some formulas holding in H(BSO n ) involving the action of the Milnor operations Q i on u 2 . These formulas have formally identical analogues in topology and we present a proof just for completeness.
Suppose Q i−1 u 3 = θ 2 i−1 . Therefore, by Cartan formula since for i ≥ 2 we have that Sq 2 i u 3 = 0 by Wu formula, which completes the proof. of H(BSO n ) for any i ≥ 1. Moreover, by Lemma 11.1 and since w 2 maps to τ u 2 via the homomorphism H(Bé t SO n ) → H(BSO n ), we deduce that τ θ 2 i = τ Sq 1 θ i+1 = Sq 1 Sq 2 i · · · Sq 1 w 2 vanishes in Chern(Bé t Spin n ) for all i ≥ 1. Proof. The ideal I n is just the kernel of the epimorphism Z /2[c 2 , . . . , c n ] → Chern(Bé t Spin n ). Then, the first inclusion of the chain is justified by Remark 11.3. The second inclusion follows from the fact that the epimorphism Z /2[c 2 , . . . , c n ] → Chern(BSpin n ) factors through Chern(Bé t Spin n ) and by Theorem 8.3.
As we have already mentioned, the previous result is analogous to [21, Corollary 5.2] (which is stated over complex numbers) but it is valid more generally without further restriction on the base field (provided that ρ = 0). This suggests that studying Nisnevich classifying spaces could also be useful for the understanding of the Chow ring ofétale classifying spaces over more general fields where one usually lacks topological insights.