Geometric Methods for Cohomological Invariants

Pierre Guillot

Received: February 2, 2007 Revised: November 8, 2007

Communicated by Alexander Merkurjev

Abstract. We explain how to exploit Rost’s theory of Chow groups with coefficients to carry some computations of cohomological invariants. In particular, we use the idea of the “stratification method” introduced by Vezzosi.

We recover a number of known results, with very different proofs. We obtain some new information on spin groups.

Keywords and Phrases: cohomological invariants, algebraic cycles.

§1. Introduction

In what follows, k is a base field, and all other fields considered in this paper will be assumed to contain k. We fix a prime number p which is different from char(k), and for any field K/k, we write H^{i}(K) for H^{i}(Gal(K_{s}/K), Z/p(i)). Here K_{s} is a separable closure of K, and Z/p(i) is the i-th Tate twist of Z/p, that is Z/p ⊗ μ_{p}^{⊗i}. (See the end of this introduction for more information on our choice of coefficients.)

1.1. Cohomological invariants

Given a functor A(−) from fields over k to the category of pointed sets, the (mod p) cohomological invariants of A are all transformations of functors

a : A(−) → H^{∗}(−).

Typical examples for A include

A(K) = isomorphism classes of nondegenerate quadratic forms over K,
A(K) = isomorphism classes of octonions algebras over K,

etc...

For more examples and a very good introduction to the subject, see [5]. We are mostly interested in the situation where A(K) is the set of isomorphism classes over K of some type of ”algebraic object”, especially when all such objects become isomorphic over algebraically closed fields. In this case, if we write X_{K} for the base point in A(K), then we obtain a group scheme G over k by setting G(K) = Aut_{K}(X_{K}), assuming Aut_{K} is appropriately defined. Moreover, by associating to each object Y_{K} ∈ A(K) the variety of isomorphisms (in a suitable sense) from Y_{K} to the base point X_{K}, we obtain a 1 : 1 correspondence between A(K) and H^{1}(K, G). Recall that this is the (pointed) set of all G-torsors over K, i.e. varieties acted on by G which become isomorphic to G with its translation action on itself when the scalars are extended to the algebraic closure K̄. For example, in the case of quadratic forms, resp. of octonion algebras, we have A(K) = H^{1}(K, O_{n}), resp. A(K) = H^{1}(K, G_{2}). In more geometric terms, a G-torsor over K is a principal G-bundle over Spec(K). In this way we see that cohomological invariants

H^{1}(−, G) → H^{∗}(−)

are analogous to characteristic classes in topology. We shall call them the mod p cohomological invariants of G. They form a graded H^{∗}(k)-algebra which we denote by Inv(G) (the prime p being implicit).

1.2. Versal torsors and classifying spaces

A simple way of constructing a G-torsor is to start with a G-principal bundle E → X and take the fibre over a point Spec(K) → X. As it turns out, there always exist bundles π : E → X with the particularly nice feature that any Gtorsor is obtained, up to isomorphism, as a fibre of π. In this case, the generic torsor T_{κ} over κ = k(X) is called versal. We shall also say that π is a versal bundle. A strong result of Rost (presented in [5]) asserts that an invariant a ∈ Inv(G) is entirely determined by its value a(T_{κ}) on a versal torsor. It follows that Inv(G) ⊂ H^{∗}(κ), and more precisely one can show that

Inv(G) ⊂ A^{0}(X, H^{∗}).

The right hand side refers to the cohomology classes in H^{∗}(κ) which are unramified at all divisors of X (see loc. cit.). (The notation comes from Rost’s theory of Chow groups with coefficients, see below and section 2.) There are two well-known constructions of versal bundles. One can embed G in a “special” group S (eg S = GL_{n}, S = SL_{n} or Sp_{n}) and take E = S, X = S/G. Alternatively, one can pick a representation V of G such that the action is free on a nonempty open subset U ⊂ V, and take E = U, X = U/G. In either situation, there are favorable cases when we actually have

Inv(G) = A^{0}(X, H^{∗}).

For versal bundles of the first type, this happens when S is simply connected (see Merkurjev [8]). For versal bundles of the second type, this happens when the complement of U has codimension ≥ 2 ([5], letter by Totaro). In this paper we shall restrict our attention exclusively to the second type of versal torsors, for reasons which we shall explain in a second. We shall view U/G as an approximation to the classifying space of G (see [13]). It will not cause any confusion, hopefully, to call this variety BG (see §2.3).

1.3. Rost’s Chow groups with coefficients

The group A^{0}(X, H^{∗}), for any variety X, is the first in a sequence of groups A^{n}(X, H^{∗}), Rost’s Chow groups with coefficients. They even form a (bigraded) ring when X is smooth. The term “Chow groups” is used since the usual mod p Chow groups of X may be recovered as CH^{n}X ⊗_{Z} Z/p = A^{n}(X, H^{0}). Rost’s groups have extremely good properties: homotopy invariance, long exact sequence associated to an open subset, and so on. There is even a spectral sequence for fibrations. Our aim in this paper, very briefly, is to use these geometric properties in order to get at A^{0}(BG, H^{∗}) = Inv(G). To achieve this, we shall be led to introduce the equivariant Chow groups A_{G}^{n}(X, H^{∗}), defined when X is acted on by the algebraic group G. The definition of these is in perfect analogy with the case of equivariant, ordinary Chow groups as in [3], [13], where “Borel constructions” are used. When X = Spec(k), we have A_{G}^{n}(Spec(k), H^{∗}) = A^{n}(BG, H^{∗}) where BG is a classifying space “of the second type” as above. The key observation for us will be that A_{G}^{0}(V, H^{∗}) = Inv(G) when V is a representation of G (thus a variety which is equivariantly homotopic to Spec(k), so to speak). This allows us to cut out V into smaller pieces, do some geometry, and eventually implement the “stratification method”, which was first introduced by Vezzosi in [14] in the context of ordinary Chow groups.

1.4. Results

A large number of the results that we shall obtain in this paper are already known, although we provide a completely different approach for these. Occasionally we refine the results and sometimes we even get something new. In this introduction we may quote the following theorem (3.2.1 in the text):

Theorem – Let p = 2 and n ≥ 2. There are exact sequences:

0 ⟶ Inv(O_{n}) ⟶ Inv(Z/2 × O_{n−1}) ⟶^{r} Inv(O_{n−2})
0 ⟶ Inv(SO_{2n}) ⟶ Inv(Z/2 × SO_{2n−1}) ⟶^{r} Inv(SO_{2n−2})
0 ⟶ Inv(Spin_{2n}) ⟶ Inv(Z/2 × Spin_{2n−1}) ⟶^{r} Inv(Spin_{2n−2})

Moreover, the image of the map r contains the image of the restriction map Inv(O_{n}) → Inv(O_{n−2}), resp. Inv(SO_{2n}) → Inv(SO_{2n−2}), resp. Inv(Spin_{2n}) → Inv(Spin_{2n−2}).

In each of the three cases the existence of the map r was not known, as far as I am aware, while the other half of the exact sequence is described by Garibaldi in [4]. Of course the computation of Inv(O_{n}) and Inv(SO_{n}) has been completed (see [5] again), and indeed we shall derive it from the theorem. This affords a new construction of an invariant first defined by Serre (and written originally b_{1}). On the other hand the invariants of Spin_{n} are not known in general: they have been computed for n ≤ 12 only ([4]), so our theorem might be of some help. In any case, our emphasis in this paper is with methods rather than specific results, and the point that we are trying to make is that the stratification method is a powerful one. It provides us with a place to start when trying to tackle the computation of Inv(G), whatever G may be. It is much more mechanical than any other approach that I am aware of. Still, the reader will find in what follows a number of examples of computations of Inv(G) for various G’s: products of copies of μ_{p} in 2.2.3, the group Spin_{7} in 4.2.2, the wreath product G_{m} ≀ Z/2 in 4.2.1, the dihedral group D_{8} in 5.2.3, etc.

1.5. Organization of the paper

We start by presenting Rost’s definition of Chow groups with coefficients in section 2. We also indicate how to construct the equivariant Chow groups, and we mention a number of basic tools such as the Künneth formula. In section 3 we introduce the stratification method, and prove the above results on orthogonal groups. We compute Inv(O_{n}) and Inv(SO_{n}) completely. In section 4 we explain how one could use the stratification method with projective representations rather than ordinary ones. We recover a result (corollary 4.1.3) which was proved and exploited fruitfully in [4]. We proceed to introduce the Bloch & Ogus spectral sequence in section 5. As we have mentioned already, the stratification method has been used by many authors (including, Vezzosi, Vistoli and myself) to compute CH^{∗}BG and H^{∗}BG, and to study the cycle map CH^{∗}BG → H^{∗}BG. It turns out that the spectral sequence shows, roughly speaking, that cohomological invariants measure the failure of this cycle map to be an isomorphism (in small degrees this is strictly true). This was the motivation to try and extend the stratification method to cohomological invariants. We conclude in section 6 with some easy remarks on invariants with values in other cycle modules, particularly algebraic K-theory. Notations & Conventions. We insist on the assumptions that we have made at the beginning of this introduction: p is a fixed prime, k is a fixed base field, p ≠ char(k), and H^{∗}(K) means mod p Galois cohomology as defined more precisely above. Our particular choice of twisting for the coefficients has been dictated by the desire to obtain a ”cycle module” in the sense described in section 2, and at the same time keep things as simple as possible. There are more general cycle modules, including Galois cohomology simply with some more general coefficients (see [11]). In most applications, we have p = 2 anyway, in which case the twisting does not interfere. We will say that G is an algebraic group to indicate that G is a smooth, affine group scheme over k [in alternative terminology: G is a linear algebraic group over k̄ defined over k]. Subgroups will always be assumed to be closed and smooth. We shall encounter many times the stabilizer of an element under an action of G: in each case, it will be trivially true that the scheme-theoretic stablizer is closed and smooth. [Alternatively: subgroups are always defined over k. So will all stabilizers we encounter.] We write Inv(G) for the mod p cohomological invariants of G, even though the letter p does not appear in the notation. We caution that in [4], these would be called the invariants of G with values in μ_{p} (for p = 2 there is no difference).

Acknowledgements. I would like to thank Burt Totaro and Skip Garibaldi for their many suggestions.

§2. Rost’s Chow groups with coefficients

2.1. Definitions

In [11], Rost defines a cycle module to be a functor M : F ↦ M(F) from fields containing k to graded abelian groups, equipped with structural data. Quoting Rost, these divide into ”the even ones”, namely restriction and corestriction maps, and ”the odd ones”: it is required that M(F) should have the structure of a module over K_{∗}(F) (Milnor’s K-theory of F), and there should exist residue maps for discrete valuations. The fundamental example for us is M(F) = H^{∗}(F) (mod p Galois cohomology as defined in the introduction). Another example is Milnor’s K-theory itself, M(F) = K_{∗}(F). Given a cycle module M, Rost defines for any variety X over k the Chow groups with coefficients in M, written A_{i}(X, M) (i ≥ 0). These are bigraded; for the cycle modules above, we have the summands A_{i}(X, H^{j}) and A_{i}(X, K_{j}). Moreover, the classical and mod p Chow groups can be recovered as CH_{i}X = A_{i}(X, K_{0}) and CH_{i}X ⊗_{Z} Z/p = A_{i}(X, H^{0}). When X is of pure dimension n, we shall reindex the Chow groups by putting A^{i}(X, H^{j}) = A_{n−i}(X, H^{j}). Our interest in the theory of Chow groups with coefficients stems from the concrete description of A^{0}(X, H^{j}) when X is irreducible: it is the subgroup of H^{j}(k(X)) comprised of those cohomology classes in degree j which are unramified at all divisors of X. Hence, when X is a classifying space for an algebraic group G (as in 1.2), we have A^{0}(X, H^{j}) = Inv^{j}(G). Rost’s Chow groups have all the properties of ordinary Chow groups, and more: in fact they were designed to be more flexible than ordinary Chow groups, particularly in fibred situations. For the time being, we shall be content to list the following list of properties; we will introduce more as we go along. Here X will denote an equidimensional variety.

1. When X is smooth, A^{∗}(X, H^{∗}) is a graded-commutative ring.

2. A map f : X → Y induces f^{∗} : A^{∗}(Y, H^{∗}) → A^{∗}(X, H^{∗}) whenever Y is smooth or f is flat.

3. A proper map f : X → Y induces f_{∗} : A_{∗}(X, H^{∗}) → A_{∗}(Y, H^{∗}).

4. There is a projection formula: f_{∗}(xf^{∗}(y)) = f_{∗}(x)y (here f : X → Y is proper, and X and Y are smooth).

5. Let i : C → X be the inclusion of a closed subvariety, and let j : U → X denote the inclusion of the open complement of C. Then there is a long exact sequence:

⋯ ⟶ A_{m}(C, H^{n}) ⟶^{i_{∗}} A_{m}(X, H^{n}) ⟶^{j^{∗}}
A_{m}(U, H^{n}) ⟶^{r} A_{m−1}(C, H^{n−1}) ⟶ ⋯

Moreover the residue map r, or connecting homomorphism, satisfies

r(xj^{∗}(y)) = r(x)i^{∗}(y).

6. If π : E → X is the projection map of a vector bundle or an affine bundle, then π^{∗} is an isomorphism.

There are particular cases when we can say more about the exact sequence in property (5). When n = 0 for instance, we recover the usual localisation sequence:

CH_{m}C ⊗_{Z} Z/p ⟶ CH_{m}X ⊗_{Z} Z/p ⟶ CH_{m}U ⊗_{Z} Z/p ⟶ 0.

Also, when X is equidimensional and C has codimension ≥ 1 (for example when X is irreducible and U is nonempty), then the following portion of the sequence is exact:

0 ⟶ A^{0}(X, H^{n}) ⟶^{j^{∗}} A^{0}(U, H^{n}).

Example 2.1.1. Let us compute A^{0}(G_{m}, H^{∗}), where G_{m} = GL_{1} is the punctured affine line. The inclusion of the origin in A^{1} yields the exact sequence:

0 ⟶ A^{0}(A^{1}, H^{∗}) ⟶ A^{0}(G_{m}, H^{∗}) ⟶
A^{0}(Spec(k), H^{∗−1}) ⟶ A^{1}(A^{1}, H^{∗−1}) .

Now, A^{1} can be seen as a (trivial) vector bundle over Spec(k), so A^{i}(A^{1}, H^{∗}) = A^{i}(Spec(k), H^{∗}). From the definitions, we see directly that A^{i}(Spec(k), H^{∗}) = 0 when i > 0, and A^{0}(Spec(k), H^{∗}) = H^{∗}(k). So we have the exact sequence of H^{∗}(k)-modules:

0 ⟶ H^{∗}(k) ⟶ A^{0}(G_{m}, H^{∗}) ⟶ H^{∗−1}(k) ⟶ 0.

It follows that A^{0}(G_{m}, H^{∗}) is a free H^{∗}(k)-module on two generators, one in degree 0, the other in degree 1. In fact, if we write k(G_{m}) = k(t), it is not difficult to see that the element (t) ∈ H^{1}(k(t)) = k(t)^{×}/(k(t)^{×})^{p} can be taken as the generator in degree 1. For this one only has to unwind the definition of r as a residue map: see [5], chap. II and III. Essentially the reason is that the divisor of the function t on A^{1} is the origin. In any case, we shall use the letter t for this generator. When k is algebraically closed, so that the ring H^{∗}(k) can be seen as the field F_{p} concentrated in degree 0, then we can write A^{0}(G_{m}, H^{∗}) = F_{p}[t]/(t^{2}), an exterior algebra.

Remark 2.1.2 – Let X be smooth, and suppose that X has a k-rational point Spec(k) → X. The induced map on Chow groups gives a splitting for the map H^{∗}(k) → A^{0}(X, H^{∗}) coming from the projection X → Spec(k). It follows that we can write

A^{0}(X, H^{∗}) = H^{∗}(k) ⊕ A^{0}(X, H^{∗})_{norm}.

The elements in the second summand are said to be normalised. Note that the splitting may a priori depend on the choice of a k-rational ”base point” for X. There is an analogous notion for cohomological invariants. Namely, an invariant n ∈ Inv(G) is called normalised when it is 0 on the trivial torsor; an invariant c ∈ Inv(G) is called constant when there is an α ∈ H^{∗}(k) such that for any torsor T over a field K, the value c(T) is the image of α under the natural map H^{∗}(k) → H^{∗}(K). Any invariant a ∈ Inv(G) may be written a = n + c with n normalised and c constant, and we may write

Inv(G) = H^{∗}(k) ⊕ Inv(G)_{norm}

where the second summand consists of normalised elements. Whenever X is a classifying space of the types considered in the introduction, there is a canonical choice of a point Spec(k) → X such that the pullback of the versal torsor over X is the trivial torsor. It follows that A^{0}(X, H^{∗})_{norm} = Inv(G)_{norm}. Let us give an application. The group μ_{p} acts freely on G_{m}, and the quotient is again G_{m}. Since G_{m} is special, it follows from the introduction that the torsor G_{m} → G_{m} is versal, so Inv(μ_{p}) ⊂ A^{0}(G_{m}, H^{∗}). It is not immediate that this inclusion is an equality, as G_{m} is not a classifying space: G_{m} is not simply-connected, and the origin has only codimension 1 in A^{1}, so we are in neither of the two favorable cases. To show that it is indeed an equality, we remark that A^{0}(G_{m}, H^{∗})_{norm} is a free H^{∗}(k)-module on one generator of degree 1, from the previous example. As a result, if we can find a non-zero, normalised invariant of μ_{p} in degree 1, then the map of H^{∗}(k)-modules Inv(μ_{p})_{norm} → A^{0}(G_{m}, H^{∗})_{norm} will certainly be surjective. Such an invariant is easy to find. Indeed, take the identity H^{1}(K, μ_{p}) → H^{1}(K). (Should you try to compute the mod p invariants of μ_{ℓ}, where ℓ is a prime different from p, you would find no normalized invariants at all. It is an easy exercise to prove this using the techniques to be introduced in this paper).

2.2. Spectral sequences and Künneth formula

Apart from Milnor K-theory and Galois cohomology, there is one extra type of cycle module which we shall consider in this paper. In the presence of a map f : X → Y, and having picked a primary cycle module M, there is for each n ≥ 0 a new cycle module written A_{n}[f; M], see [11],§7. These are functors from fields κ ”over Y ”, that is fields with a map Spec(κ) → Y, to graded abelian groups – which turns out to be enough to define A_{∗}(Y, A_{n}[f; M]). Quite simply, if we write X_{κ} = Spec(κ) ×_{Y} X, the definition is:

A_{n}[f; M](κ) = A_{n}(X_{κ}, M).

When f and M are understood, we shall write A_{n} for A_{n}[f; M]. Also, when f is flat, so that the fibres all have the same dimension d, we may change the grading to follow the codimension, thus defining A^{n} = A_{d−n}.

Theorem 2.2.1 – Let f : X → Y be a flat map, and suppose that Y is equidimensional. Let A_{n} = A_{n}[f; M] for a cycle module M. Then there exists a convergent spectral sequence:

A^{r}(Y, A^{s}) ⇒ A^{r+s}(X, M).

See [11],§8, for a proof.

Corollary 2.2.2 – Let X be a scheme (over k), and write X_{κ} for the scheme obtained by extending the scalars to the field κ. Assume that A^{0}(X, H^{∗}) is a free H^{∗}(k)-module of finite rank, and that

A^{0}(X_{κ}, H^{∗}) = A^{0}(X, H^{∗}) ⊗_{H^{∗}(k)} H^{∗}(κ).

Then there is a Künneth isomorphism:

A^{0}(X × Y, H^{∗}) = A^{0}(X, H^{∗}) ⊗_{H^{∗}(k)} A^{0}(Y, H^{∗})

for any equidimensional Y.

The hypotheses of this corollary should be compared with [5], 16.5.

Proof. Let f : X × Y → Y be the projection. By the theorem, we know that A^{0}(X × Y, H^{∗}) = A^{0}(Y, A^{0}). However, for any field Spec(κ) → Y, the fibre of f above Spec(k) is X_{κ} as in the statement of the corollary. Our hypothesis is then that A^{0}(κ) = A^{0}(X_{κ}, H^{∗}) is a direct sum of copies of H^{∗}(κ), and more precisely that A^{0} splits up as the direct sum of several copies of the cycle module H^{∗}. The result follows.

As an illustration, we have the following proposition.

Proposition 2.2.3 – There are invariants t^{(i)} ∈ Inv((μ_{p})^{n}) for 1 ≤ i ≤ n, each of degree 1, such that Inv((μ_{p})^{n}) is a free H^{∗}(k)-module on the products t^{(i_{1})}t^{(i_{2})} ⋯ t^{(i_{k})}, for all sequences 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ n. When k is algebraically closed, Inv((Z/p)^{n}) is an exterior algebra Λ(t^{(1)}, …, t^{(n)}) over F_{p}.

Proof. The case n = 1 has been dealt with in remark 2.1.2. The general case follows from corollary 2.2.2.

2.3. Equivariant Chow groups

The good properties of Chow groups with coefficients will allow us to define the equivariant Chow groups of a variety acted on by an algebraic group. This will be in strict analogy with [3] and [13], to which we will refer for details. So let G be a linear algebraic group over k. There exists a representation V of G such that the action is free outside of a closed subset S; moreover, we can make our choices so that the codimension of S is arbitrarily large (loc. cit.). Put U = V − S. Let X be any equidimensional scheme over k with an action of G. Define

X_{G} = \frac{U × X}{G} .

We shall restrict our attention to those pairs (X, U) for which X_{G} is a scheme rather than just an algebraic space. Lemma 9 and Proposition 23 in [3] provide simple conditions on X under which an appropriate U may be found. This will be amply sufficient for the examples that we need to study in this paper.^{1} The notation X_{G} hides the dependence on U because, as it turns out, the Chow groups A^{i}(X_{G}, H^{∗}) do not depend on the choice of V or S, as long as the codimension of S is large enough (depending on i). One may prove this using the ”double fibration argument” as in [13]. We write A_{G}^{i}(X, H^{∗}) for this group (or the limit taken over all good pairs (V, S), if you want). A map f : X → Y induces a map f_{G} : X_{G} → Y_{G} (the notation X_{G} will mean that a choice of V and S has been made). It follows that the equivariant Chow groups with coefficients A_{G}^{∗}(−, H^{∗}) are functorial, and indeed they have all the properties listed in §2.1. The proof of this uses that if f is flat, proper, an open immersion, a vector bundle projection, etc, then f_{G} is respectively flat, proper, an open immsersion, or a vector bundle projection (see [3]). It may be useful to spell out that a G-invariant open subset U in an equidimensional G-variety X, whose complement C has codimension c, gives rise to a long exact sequence:

⋯ → A_{G}^{m}(C, H^{n}) → A_{G}^{m+c}(X, H^{n}) → A_{G}^{m+c}(U, H^{n}) → A_{G}^{m+1}(C, H^{n−1}) → · · · .

We shall refer to it as the equivariant long exact sequence associated to U.

Example 2.3.1. We shall write BG for Spec(k)_{G}. It follows from the results above and from the introduction that

A_{G}^{0}(Spec(k), H^{∗}) = A^{0}(BG, H^{∗}) = Inv(G).

1I am grateful to the referee for pointing out this technical difficulty. I also agree with him or her that it would be desirable to extend Rost’s theory of Chow groups with coefficients to algebraic spaces. This is certainly not the place to do so.

Documenta Mathematica 12 (2007) 521–545

Example 2.3.2. Suppose that the action of G on X is free, and that the quotient X/G exists as a scheme. Then there is a natural map X_{G} → X/G. Moreover X_{G} is an open subset in (V ×X)/G, which in turn is (the total space of) a vector bundle over X/G. The complement of X_{G} in (V × X)/G can be taken to have an arbitrarily large codimension (namely, it is that of S). In a given degree i, we may thus choose V and S appropriately and obtain:

A_{G}^{i}(X, H^{∗}) = A^{i}(X_{G}, H^{∗}) = A^{i}((V × X)/G, H^{∗}) = A^{i}(X/G, H^{∗}).

(The second equality coming from the long exact sequence as in §2.1, property (5).) Thus when the action is free, the equivariant Chow groups are just the Chow groups of the quotient.

Notations 2.3.3 – As we have already done in this section, we shall write X_{G} to signify that we have chosen V and S with the codimension of S large enough for the computation at hand. Thus we can and will write indifferently A_{G}^{∗}(X, H^{∗}) or A^{∗}(X_{G}, H^{∗}). Moreover, we shall write EG for U = V − S, with the same convention. The quotient EG/G is written BG, and we write A^{∗}(BG, H^{∗}) much more often than A_{G}^{∗}(Spec(k), H^{∗}).

Remark 2.3.4 – Whenever we have a theory at hand which has the properties listed in §2.1, we can define the equivariant analog exactly as above. Apart from Chow groups with coefficients, we shall have to consider at one point the étale cohomology of schemes, namely the groups H_{et}^{i}(X, Z/p(i)), which we shall write simply H_{et}^{i}(X), the coefficients being understood. In fact we shall only encounter the expression H_{et}^{i}(BG). When k = C, we have H_{et}^{∗}(BG) = H^{∗}(BG, F_{p}), where on the right hand side we use topological cohomology and a model for the classifying space BG in the classical, topological sense of the word. To see this, note first that H_{et}^{∗}((V − S)/G) = H^{∗}((V − S)/G, F_{p}) since we are using finite coefficients. Moreover, we can arrange V and S so that V − S is a Stiefel variety (see [13]), and this provides sufficiently many V ’s and S’s (that is, the codimension of S can be arbitrarily large, with V − S a Stiefel variety). As a result, we can restrict attention to these particular pairs (V, S) when forming the limit H_{et}^{∗}(BG) = lim H^{∗}((V − S)/G, F_{p}) without changing the result, which is then clearly H^{∗}(E/G, F_{p}) where E is the infinite Stiefel variety. This space E is contractible, so E/G is a topological model for BG.

§3. The stratification method

3.1. The idea of the method

Let G be a linear algebraic group. The stratification method is a procedure to compute A^{0}(BG, H^{∗}) = Inv(G). We shall not try to present it as a mechanical algorithm, but rather as a heuristic recipe. This being said, one of the virtues of the method is that it is closer to a systematic treatment of the question than any other approach that the author is aware of. The stratification method rests on two very elementary facts.

1. If K is a closed subgroup of G, then the equivariant Chow groups of the variety G/K have an easy description: indeed there is an isomorphism of varieties:

(G/K)_{G} = \frac{EG × G/K}{G} = \frac{EG}{K} = BK.

Thus A_{G}^{∗}(G/K, H^{∗}) = A^{∗}(BK, H^{∗}).

2. Suppose that V is a representation of G. Then it is a G-equivariant vector bundle over a point, and therefore V_{G} → Spec(k)_{G} = BG is a vector bundle, too. As a result, A_{G}^{∗}(V, H^{∗}) = A^{∗}(BG, H^{∗}).

Things are put together in the following way. One starts with a well-chosen representation V, and then cuts V into smaller pieces. These smaller pieces would ideally be orbits or families of orbits parametrized in a simple manner. Using (1), and hoping that the stabilizer groups K showing up are easy enough to understand, one computes the Chow groups of the small pieces. Applying then repeatedly the equivariant long exact sequence from §2.3, one gets at A_{G}^{0}(V, H^{∗}). By (2), this is A^{0}(BG, H^{∗}). Any particular application of the method will require the use of ad hoc geometric arguments. Before giving an example however, we propose to add a couple of lemmas to our toolkit.

Lemma 3.1.1 – Suppose that G and K are algebraic groups acting on the left on X, and suppose that the actions commute. Assume further that the action of K alone gives a K-principal bundle X → X/K = Y. Then

A_{G}^{∗}(Y, H^{∗}) = A_{G×K}^{∗}(X, H^{∗}).

Proof. This is simply an elaboration on example 2.3.2.

We shall have very often the occasion of using this lemma in the following guise:

Lemma 3.1.2 – Suppose that G contains a subgroup G′ which is an extension

1 → N → G′ → K → 1.

Let G × K act on G/N by (σ, τ) · [g] = [σgτ^{−1}]. Assume moreover that K acts on a variety X, and that there is a K-principal bundle G/N × X → Y. Then

A_{G}^{∗}(Y, H^{∗}) = A_{G′}^{∗}(X, H^{∗}).

If the product N × K is direct, we have X_{N} × K = BN × X_{K}.

Here the variety Y is an example of what we call ”a family of orbits (of G) parametrized in a simple manner”. Note that when K is the trivial group and X = Spec(k), we recover property (1) above.

Proof. By lemma 3.1.1, we have

A_{G}^{∗}(Y, H^{∗}) = A_{G×K}^{∗}(G/N × X, H^{∗}).

Now let E_{1} be an open subset in a representation of G, on which the action is free, and suppose that the codimension of the complement of E_{1} is large enough. Similarly, pick E_{2} for K; then E_{1} × E_{2} can be chosen for the group G×K. We regard E_{1} as a trivial K-space and E_{2} as a trivial G-space. Finally, G′ has natural maps to both G and K, and we combine these to see E_{1} × E_{2} as a G′-space, with non-trivial action on each factor. We write:

\begin{aligned}G/N × X_{G} × K & = & (E_{1} × E_{2} × \frac{G}{N} × X)/G × K \\ & = & (\frac{E_{1}×G/N}{G} × E_{2} × X)/K \end{aligned} .

By property (1) above, we may identify \frac{E_{1}×G/N}{G} with E_{1}/N = BN. Moreover we have arranged things so that, under this identification, the action of K on \frac{E_{1}×G/N}{G} translates into its natural action on E_{1}/N (that is, the action induced from that which G′ possesses as a subgroup of G normalising N). We proceed:

\begin{aligned}G/N × X_{G} × K & = & (E_{1}/N × E_{2} × X)/K \\ & = & (E_{1} × E_{2} × X)/G′ \end{aligned} .

When the product is direct, the action of K on E_{1}/N is trivial. This concludes the proof.

As an example, we may apply this to actions of G on disjoint unions:

Lemma 3.1.3 – Suppose that G acts on Y and that Y is the disjoint union of varieties Y_{1} ∐ … ∐ Y_{n}. Let X = Y_{1}, and suppose that X contains a point in each orbit, that is, suppose that the natural map

f : G × X → Y

is surjective. Let K be the subgroup of G leaving X invariant. Then there is an identification:

A_{G}^{0}(Y, H^{∗}) = A_{K}^{0}(X, H^{∗}).

Proof. This is immediate from the previous lemma, with N the trivial group.

3.2. The example of O_{n}, SO_{n}, and Spin_{n}

For this example we take p = 2 (so that char(k) ≠ 2). Let q be a non-degenerate quadratic form on a k-vector space V of dimension n. Assume moreover that q has maximal Witt index. Then its automorphism group, which we will denote by O_{n}, is split. Similarly one has the groups SO_{n} and Spin_{n}, which are also split.

Theorem 3.2.1 – Let p = 2 and n ≥ 2. There are exact sequences:

0 ⟶ Inv(O_{n}) ⟶ Inv(Z/2 × O_{n−1}) ⟶^{r} Inv(O_{n−2})
0 ⟶ Inv(SO_{2n}) ⟶ Inv(Z/2 × SO_{2n−1}) ⟶^{r} Inv(SO_{2n−2})
0 ⟶ Inv(Spin_{2n}) ⟶ Inv(Z/2 × Spin_{2n−1}) ⟶^{r} Inv(Spin_{2n−2})

Moreover, the image of the map r contains the image of the restriction map Inv(O_{n}) → Inv(O_{n−2}), resp. Inv(SO_{2n}) → Inv(SO_{2n−2}), resp. Inv(Spin_{2n}) → Inv(Spin_{2n−2}).

Proof. Step 1. Let G_{n} denote either of O_{n}, SO_{n}, or Spin_{n}. Then G_{n} has the canonical representation V. Let V′ denote V − {0}. We write U for the open subset in V′ on which q is non-zero, and we write C for its complement. The codimension of {0} in V is n. Consider the long exact sequence:

0 → A_{G_{n}}^{0}(V, H^{∗}) → A_{G_{n}}^{0}(V′, H^{∗}) →

0 = A_{G_{n}}^{1−n}({0}, H^{∗−1}) → A_{G_{n}}^{1}(V, H^{∗−1}) → A_{G_{n}}^{1}(V′, H^{∗−1}).

It follows that A_{G_{n}}^{0}(V, H^{∗}) = A_{G_{n}}^{0}(V′, H^{∗}) = Inv(G_{n}). Moreover the last map above is surjective when ∗ = 1 (cf §2.1), so that A_{G_{n}}^{1}(V, H^{0}) = A_{G_{n}}^{1}(V′, H^{0}) = CH^{1}BG_{n} ⊗_{Z} Z/2. Turning to the equivariant long exact sequence associated to the open set U in V′, we finally get:

0 ⟶ Inv(G_{n}) ⟶ A_{G_{n}}^{0}(U, H^{∗}) ⟶ A_{G_{n}}^{0}(C, H^{∗−1}).  (†)

When ∗ = 1 we have:

0 → Inv^{1}(G_{n}) → A_{G_{n}}^{0}(U, H^{1}) →

A^{0}(C, H^{0}) → CH^{1}BG_{n} ⊗_{Z} Z/2 → CH_{G_{n}}^{1}U ⊗_{Z} Z/2 → 0. (††)

Step 2. Let Q = q^{−1}(1). We shall use the following result.

Lemma 3.2.2 – The action of G_{n} on Q is transitive. Moreover the stabilizer of a k-rational point is isomorphic to G_{n−1}, and we get an isomorphism of k-varieties Q = G_{n}/G_{n−1}. The action of G_{n} on C is transitive. Moreover the stabilizer of a k-rational point is isomorphic to a semi-direct product H ⋊G_{n−2}, where H is an algebraic group isomorphic to affine space as a variety, and we get an isomorphism of k-varieties C = G_{n}/H ⋊ G_{n−2}. Finally, the map B(H ⋊ G_{n−2}) → BG_{n−2} is an affine bundle.

For a proof, see [10]. From this we can at once identify the last term in the exact sequence (†). Indeed, if S = H ⋊ G_{n−2} as in the lemma, then A_{G_{n}}^{0}(C, H^{∗}) = A^{0}(BS, H^{∗}) as explained at the beginning of this §. However, since BS → BG_{n−2} is an affine bundle, we draw from §2.1, property (6), that A^{0}(BS, H^{∗}) = A^{0}(BG_{n−2}, H^{∗}) = Inv(G_{n−2}). Step 3. We turn to the term A_{G_{n}}^{0}(U, H^{∗}). Extend the action of G_{n} on Q to an action on Q×G_{m} which is trivial on the second factor. The group Z/2 = {1, τ} also acts on Q × G_{m} by τ(x, t) = (−x, −t). The two actions commute. Finally, there is a map Q × G_{m} → U defined by (x, t) ↦ tx. It is G_{n}-equivariant, and a Z/2-principal bundle. We wish to apply lemma 3.1.2 with N = G_{n−1}, K = Z/2, and G′ = N × K. This is of course the time when we need to assume that n is even if G_{n} is SO_{n} or Spin_{n}. Then the element −Id generates the copy of Z/2 that we need. We conclude:

A_{G_{n}}^{0}(U, H^{∗}) = A^{0}(BG_{n−1} × (G_{m})_{Z/2}, H^{∗}).

We may easily compute A^{0}((G_{m})_{Z/2}, H^{∗}). Indeed, using that the action is free, with quotient G_{m}, this is A^{0}(G_{m}, H^{∗}) from example 2.3.2. As observed in remark 2.1.2, this happens to equal A^{0}(BZ/2, H^{∗}) = Inv(Z/2). Now, by the Künneth formula (corollary 2.2.2), which we may use as Inv(Z/2) is a free H^{∗}(k)-module, we draw

A^{0}(BG_{n−1}×(G_{m})_{Z/2}, H^{∗}) = Inv(G_{n−1})⊗_{H^{∗}(k)}Inv(Z/2) = Inv(Z/2×G_{n−1}).

Step 4. We need to prove the last sentence in the theorem. We claim that there is an x ∈ Inv(Z/2 × G_{n−1}) such that r(x) = 1 ∈ Inv(G_{n−2}). Granted this, the theorem follows from the formula r(xj^{∗}(y)) = r(x)i^{∗}(y) = i^{∗}(y) (see §2.1), applied here with i^{∗} denoting the restriction Inv(G_{n}) → Inv(G_{n−2}) and j^{∗} denoting the restriction Inv(G_{n}) → Inv(Z/2 × G_{n−1}). To prove the claim we start by computing CH_{G_{n}}^{1}U (in the rest of this proof we shall write simply CH for the mod 2 Chow groups). As above this is CH^{1}BG_{n−1} × (G_{m})_{Z/2}. Arguing as in example 2.3.2, we see that we may replace (G_{m})_{Z/2} by G_{m}/Z/2 = G_{m}, as far as computing the Chow groups goes. Now, G_{m} being an open set in affine space, it has a trivial Chow ring and there is a Künneth formula, so:

CH^{1}BG_{n−1} × (G_{m})_{Z/2} = CH^{1}BG_{n−1}.

Therefore the last map in (††) is the restriction map CH^{1}BG_{n} → CH^{1}BG_{n−1}. When G_{n} = Spin_{n}, which is simply-connected, we have CH^{1}BG_{n} = 0. In [10], the reader will find a proof that CH^{1}BSO_{2n} = 0 also and that CH^{1}BO_{n} → CH^{1}BO_{n−1} is an isomorphism. The claim follows from the exact sequence (††).

3.3. End of the computation for O_{n}

The main novelty in theorem 3.2.1 is with the spin groups (specifically, with the second half of the exact sequence presented, as the first half was obtained by Garibaldi in [4]). The invariants of O_{n} and SO_{n} are completely known, see [5]. In this section and the next, however, we show how to recover these results from theorem 3.2.1. In the case of O_{n} this is so close to the proof given in loc. cit.that we shall omit most of the details. In the case of SO_{n} on the other hand, our method is considerably different, so we have thought it worthwhile to present it. We continue with p = 2. A major role will be played by Stiefel-Whitney classes. Recall that H^{1}(K, O_{n}) is the set of isomorphism classes of non-degenerate quadratic forms on K^{n}. Given such a form q, the classes w_{i}(q) ∈ H^{∗}(K), for i ≥ 0, have been defined by Milnor [9] (originally they were defined in mod 2 Milnor K-theory). They are 0 for i > n and if w_{t}(q) = ∑ w_{i}(q)t^{i}, then one has

w_{t}(q ⊕ q′) = w_{t}(q)w_{t}(q′).

Each w_{i} can be seen as a cohomological invariant in Inv(O_{n}). In fact one has

Proposition 3.3.1 – The H^{∗}(k)-module Inv(O_{n}) is free with a basis given by the classes w_{i}, for 0 ≤ i ≤ n. Moreover, the following multiplicative formula holds:

\begin{aligned}w_{1}w_{i−1} & = & w_{i} & (i odd), \\ & = & (−1)w_{i−1} & (i even). \end{aligned}

In the formula, (−1) stands for the image of −1 ∈ k in H^{1}(k) = k^{×}/(k^{×})^{2}.

Sketch proof. From theorem 3.2.1, we have an injection

0 → Inv(O_{n}) → Inv(Z/2 × O_{n−1})

where Z/2×O_{n−1} is seen as a subgroup of O_{n}, the nonzero element in the copy of Z/2 corresponding to the −Id matrix. An immediate induction then shows that Inv(O_{n}) injects into Inv((Z/2)^{n}), the invariants of the elementary abelian 2-group comprised of the diagonal matrices in O_{n} with ±1 as entries. (This could be seen as a consequence of the surjection H^{1}(K, (Z/2)^{n}) → H^{1}(K, O_{n}), which in turns expresses the fact that quadratic forms may be diagonalized). We have computed Inv(Z/2) in remark 2.1.2, and we may apply corollary 2.2.2 to obtain Inv((Z/2)^{n}). The symmetric group S_{n} acts on this ring, and the image of Inv(O_{n}) must certainly lie in the subring fixed by this action. The latter is easily seen to be a free H^{∗}(k)-module on the images of the Stiefel-Whitney classes. The multiplicative formula is obtained by direct computation.

3.4. End of the computation for SO_{n}

Since SO_{n} is a subgroup of O_{n}, the restriction map Inv(O_{n}) → Inv(SO_{n}) defines Stiefel-Whitney classes for SO_{n}. Now, as we have defined it, the group SO_{n} has the following interpretation for its torsors: H^{1}(K, SO_{n}) is the set of isomorphism classes of non-degenerate quadratic forms on K^{n} whose discriminant is 1. Recall that the discriminant is the determinant of any matrix representing the bilinear form associated to q, viewed as an element of K^{×}/(K^{×})^{2}. As it turns out, the discriminant of q is precisely w_{1}(q) when seen as a class in the additive group H^{1}(K). So we have w_{1} = 0 in Inv(SO_{n}). From the formula w_{1}w_{2i} = w_{2i+1}, we have in fact w_{2i+1} = 0 in Inv(SO_{n}).

Proposition 3.4.1 – When n = 2m + 1 is odd, Inv(SO_{n}) is a free H^{∗}(k)- module with basis {w_{0}, w_{2}, w_{4}, …, w_{2m}}. When n = 2m is even, Inv(SO_{n}) is a free H^{∗}(k)-module with basis

{w_{0}, w_{2}, w_{4}, …, w_{2m−2}, b_{n−1}^{(1)}},

where b_{n−1}^{(1)} is an invariant of degree n − 1.

The invariant b_{n−1}^{(1)} was introduced by Serre [5]. It was originally denoted by b_{1}, but we try to keep lowerscripts for the degree whenever possible.

Proof. The odd case is easy, for we have an isomorphism

Z/2 × SO_{2m+1} = O_{2m+1}.

From the Künneth formula (corollary 2.2.2), it follows that Inv(SO_{2m+1}) is the quotient of Inv(O_{2m+1}) by the ideal generated by w_{1}. From the formula in proposition 3.3.1, this is the submodule generated by the odd Stiefel-Whitney classes. The result follows. Turning to the even case, we use the exact sequence from theorem 3.2.1:

0 ⟶ Inv(SO_{2m}) ⟶ Inv(Z/2 × SO_{2m−1}) ⟶^{r} Inv(SO_{2m−2})

We proceed by induction, assuming the result for n − 2. The group Z/2 × SO_{2m−1} is seen as a subgroup of SO_{2m}, were again the copy of Z/2 is identified with {±Id}. However, there is also a canonical isomorphism of Z/2 × SO_{2m−1} with O_{2m−1}, as above. The corresponding injective map O_{2m−1} → SO_{2m} thus obtained induces the map H^{1}(K, O_{2m−1}) → H^{1}(K, SO_{2m}) which sends a quadratic form q to q̃ = (q ⊗ det(q)) ⊕ det(q). Here det(q) is the 1-dimensional quadratic form with corresponding 1 × 1 matrix given by the discriminant of q. We have w_{2i+1}(q̃) = 0 as explained above. To compute the even Stiefel-Whitney classes, we note that w_{i}(q̃) = w_{i}((q ⊕ ⟨1⟩) ⊗ det(q)) = w_{i}(q ⊗ det(q)), and we recall that if we factorise formally

∑ w_{i}(q)t^{i} = ∏(1 + a_{i}t^{i}),

then

∑ w_{i}(q ⊗ det(q))t^{i} = ∏(1 + (a_{i} + w_{1}(q))t^{i}).

It follows that w_{2i}(q̃) = w_{2i}(q)+w_{1}(q)R_{i}(q), and the remainder w_{1}(q)R_{i}(q) can thus be written as a linear combination over H^{∗}(k) of the classes w_{j}(q) with j < 2i. Moreover w_{2m}(q̃) = 0.

Now, write w_{i}, resp w′_{i}, resp w′′_{i}, for the Stiefel-Whitney classes in Inv(SO_{2m}), resp Inv(O_{2m−1}), resp Inv(SO_{2m−2}), and regard Inv(SO_{2m}) as a subring of Inv(O_{2m−1}). Thus we have w_{2i+1} = 0 and w_{2i} = w′_{2i} + w′_{1}R_{i}. Let M, resp N, denote the H^{∗}(k)-submodule of Inv(O_{2m−1}) generated by the classes w′_{2i+1} for 0 ≤ i ≤ m − 2, resp by the classes w_{2i} for 0 ≤ i ≤ m − 1 together with b_{n−1}^{(1)} = w′_{2m+1} + (−1)w′_{1}R_{m}. Then M and N are free modules, and Inv(O_{2m−1}) = M ⊕ N. We claim (i) that the residue map r is 0 on N (ie r(b_{n−1}^{(1)}) = 0), and (ii) that r maps M injectively onto a free submodule in Inv(SO_{2m−2}). This will complete the induction step, as we will have Inv(SO_{2m}) = ker(r) = N. Both parts of the claim are proved by the same computation. We first note from theorem 3.2.1 that r(w′_{1}) = 1, and we compute:

\begin{aligned}r(w′_{2i+1}) & = & r(w′_{1}w′_{2i}) \\ & = & r(w′_{1}w_{2i}) − r((w′_{1})^{2}R′_{i}) \\ & = & w′′_{2i} − r((−1)w′_{1}R_{i}) \\ & = & w′′_{2i} − (−1)r(w′_{1}R_{i}) \end{aligned} .

Take i = m − 1 to obtain (i) (since w′′_{2m−2} = 0). Property (ii) is clear. It remains to start the induction. The group SO_{2} is a torus, so Inv(SO_{2}) = H^{∗}(k). One gets the result for SO_{4} from this and theorem 3.2.1. This is very similar to the induction step (but easier), and will be left to the reader.

§4. Projective variants

4.1. Projective bundles

Consider the n-th projective space P^{n}. It has an open subset isomorphic to A^{n}, and from the long exact sequence of §2.1, (5), we draw at once A^{0}(P^{n}, H^{∗}) = H^{∗}(k). Now, if V → X is a vector bundle, we may form the associated projective bundle π : P(V) → X. The fibre of π over Spec(κ) → X is P_{κ}^{n−1}, where n is the rank of V. Thus we see that the induced cycle module A^{0} = A^{0}[π, H^{∗}] on X is isomorphic to H^{∗}, and from theorem 2.2.1 we have:

Lemma 4.1.1 – Let π : P(V) → X denote the projective bundle associated to the vector bundle V → X. Then

A^{0}(P(V), H^{∗}) = A^{0}(X, H^{∗}).

Remark 4.1.2 – It should be kept in mind that the above argument, which rests on the spectral sequence of theorem 2.2.1, has been used for the sake of concision only. The result is a consequence of the following general statement:

A^{∗}(P(V), H^{∗}) = A^{∗}(X, H^{∗})[ζ]/(ζ^{n})

with ζ ∈ A^{1}(P(V), H^{0}). This is perfectly analogous to the usual statement for ordinary Chow groups, and is no harder to prove. We shall have no use for the complete statement in the sequel, however, and therefore we omit the lengthy argument.

Corollary 4.1.3 – Let V be a representation of the linear algebraic group G. Form the projective representation P(V), and assume that there is a k-rational point in P(V) whose orbit is open, and isomorphic to G/S. Then there is an injection

0 → Inv(G) → Inv(S).

Proof. We have a vector bundle V_{G} → BG and the associated projective bundle is P(V)_{G} → BG. From the lemma,

Inv(G) = A^{0}(BG, H^{∗}) = A^{0}(P(V)_{G}, H^{∗}) = A_{G}^{0}(P(V), H^{∗}).

We may view G/S as an open subset in P(V), and from the equivariant long exact sequence we have

0 → Inv(G) → A_{G}^{0}(G/S, H^{∗}).

As noted in §3.1, we have A_{G}^{0}(G/S, H^{∗}) = Inv(S).

4.2. Applications

Example 4.2.1. As a very simple illustration, we may compute the invariants of a wreath product G_{m} ≀ Z/2, that is a semi-direct product G_{m}^{2} ⋊ Z/2 where Z/2 permutes the two copies of the multiplicative group. Indeed, this group may be seen as the normalizer of a maximal torus in GL_{2}, and thus it has a canonical representation W of dimension 2. The space P(W) is just two orbits, and that of [1, 1] is open with stabilizer G_{m} × Z/2. From corollary 4.1.3 (and corollary 2.2.2), we know that the restriction map from Inv(G_{m}^{2} ⋊ Z/2) to Inv(Z/2) is injective. It is also certainly surjective, since the inclusion Z/2 → G_{m}^{2}⋊Z/2 is a section for the projection G_{m}^{2}⋊Z/2 → Z/2. Finally, we conclude that Inv(G_{m}^{2} ⋊ Z/2) = Inv(Z/2) (regardless of the choice of p).

Example 4.2.2. In [4], Garibaldi gives many examples of applications of corollary 4.1.3. It is possible to recover a good number of his results using the techniques of this paper. Let us illustrates this with Spin_{7} at p = 2. If Δ denotes the spin representation of Spin_{7}, one can show that there exists an open orbit U in P(Δ) with stabilizer G_{2} ×Z/2 (G_{2} is the split group of that type). Hence an injection

0 → Inv(Spin_{7}) → Inv(G_{2} × Z/2).

Garibaldi computes the image of this, creating invariants of Spin_{7} by restricting invariants of Spin_{8}, and exploiting the fact that further restriction from Inv(Spin_{7}) to Inv(Z/2) is zero, since it factors through the invariants of a maximal torus.

Our version of this computation is to study the complement of U in P(Δ). It consists of a single orbit with stabilizer K, and there is an affine bundle map BK → B(SL_{3} × Z/2). Therefore we have an exact sequence

0 → Inv^{d}(Spin_{7}) → Inv^{d}(G_{2} × Z/2) → Inv^{d−1}(SL_{3} × Z/2).

This seems to be typical of the stratification method when projective representations are used: one can obtain some results very rapidly, but they are not always as accurate as one may wish. Instead, if we proceed exactly as in the proof of theorem 3.2.1, we obtain:

0 → Inv^{d}(Spin_{7}) → Inv^{d}(G_{2} × Z/2) → Inv^{d−1}(SL_{3}) → 0.

Since SL_{3} is special, Inv(SL_{3}) = H^{∗}(k). From [5] we have that Inv(G_{2}) is a free H^{∗}(k)-module on two generators of degree 0 and 3. By corollary 2.2.2, Inv(G_{2} × Z/2) is a free H^{∗}(k)-modules with 4 generators of degree 0, 1, 3 and 4. It follows that Inv(Spin_{7}) is a free H^{∗}(k)-module on 3 generators of degree 0, 3 and 4.

§5. The Bloch & Ogus spectral sequence

5.1. The spectral sequence

In this section, we assume that k is algebraically closed. The prime p being fixed as always, we shall write H_{et}^{i}(X) for the étale cohomology group H_{et}^{i}(X, Z/p).

Theorem 5.1.1 (Rost, Bloch, Ogus) – Let k = k̄, and let X be equidimensional. Then there is a spectral sequence

E_{2}^{r,s} = A^{r}(X, H^{s−r}) ⇒ H_{et}^{r+s}(X).

In particular the E_{2} page is zero under the first diagonal, and the resulting map

A^{n}(X, H^{0}) = CH^{n}X ⊗_{Z} Z/p → H_{et}^{n}(X)

is the usual cycle map.

A word of explanation on the authorship of the spectral sequence. To start with, there is the well-known coniveau spectral sequence, which converges to H_{et}^{n}(X) and for which there is a description of the E_{1} term. In [2] Bloch and Ogus prove that the E_{2} term can be identified with H_{Zar}^{r}(X, H^{s}), where H^{s} is the sheafification of U ↦ H_{et}^{s}(U). They deduce that the groups are 0 under the diagonal, and prove the statement about the cycle map. More than 20 years later, in [11], corollary 6.5, Rost proves that the E_{2} term can also be described using his Chow groups with coefficients as in the theorem. The sequence seems to be usually refered to as the Bloch & Ogus spectral sequence. Given an algebraic group G, we may take a model for BG to play the role of X, and obtain:

Corollary 5.1.2 – There is a map

H_{et}^{∗}(BG) → Inv(G)

which vanishes (when ∗ > 0) on the image of the cycle map

CH^{∗}BG ⊗_{Z} Z/p → H_{et}^{2∗}(BG).

Example 5.1.3. The Stiefel-Whitney classes as in 3.3 can be defined as the images of the elements with the same name in the cohomology of BO_{n}. The corollary explains why they square to zero in Inv(O_{n}), since they square to Chern classes in cohomology, and these come from the Chow ring (see [13] for details).

Remark 5.1.4 – When G is a finite group, viewed as a 0-dimensional algebraic group, there is a natural map

H^{∗}(G, F_{p}) → H_{et}^{∗}(BG),

where H^{∗}(G, F_{p}) is the usual cohomology of G as a discrete group. Indeed, there is a Galois covering EG → BG with group G, so we may see this map as coming from the corresponding Hoschild-Serre spectral sequence. Since we assume in this section that k is algebraically closed, it follows easily that the map is an isomorphism. Composing this with the map from the previous corollary, we obtain the homomorphism

H^{∗}(G, F_{p}) → Inv(G)

which was considered in [5]. Of course the direct definition given in loc. cit. is much preferable.

Corollary 5.1.5 (to theorem 5.1.1) – There are natural isomorphisms:

Inv^{1}(G) = H_{et}^{1}(BG)

and

Inv^{2}(G) = \frac{H_{et}^{2}(BG)}{CH^{1}BG ⊗_{Z} Z/p}.

The denominator in the second isomorphism is really the image of CH^{1}BG ⊗_{Z} Z/p in étale cohomology via the cycle map. When k = C, the cycle map is injective in degree 1 and in fact CH^{1}BG = H^{2}(BG, Z) (topological cohomology), see [13].

Corollary 5.1.6 (to theorem 5.1.1) – Any class x in the kernel of the cycle map

CH^{2}BG ⊗_{Z} Z/p → H_{et}^{4}(BG)

determines an invariant r_{x} ∈ Inv^{3}(BG), which is well-defined up to the image of H_{et}^{3}(BG) → Inv^{3}(BG). If x is nonzero, neither is r_{x}.

We think of r_{x} as a simplified version of the Rost invariant.

Proof. Our assumption implies that the class x, viewed as an element of bidegree (2, 2) on the E_{2} page of the spectral sequence under discussion, must be hit by a differential. Let r_{x} ∈ E_{2}^{0,3} = Inv^{3}(G) be such that d_{2}(r_{x}) = x. This element is well-defined up to the kernel of d_{2}. Since further differentials d_{r} for r > 2 are zero on E_{r}^{0,3}, we see that r_{x} is defined up to elements which survive to the E_{∞} page. These are, by definition, the elements in the image of H_{et}^{3}(BG) → Inv^{3}(BG).

5.2. Applications

For simplicity, we shall take k = C, the complex numbers, in the applications. In this case according to remark 2.3.4, the étale cohomology of BG as above coincides with the topological cohomology of a topological classifying space (i.e. a quotient EG/G of a contractible space EG endowed with a free G-action). We start with a proposition which should be compared with statements 31.15 and 31.20 in [7], for which there is no proof available as far as I am aware.

Proposition 5.2.1 – Over the complex numbers, there are isomorphisms

Inv^{1}(G) = Hom(π_{0}(G), Z/p)

and

Inv^{2}(G) = p-torsion in H^{3}(BG, Z).

In particular, if G is connected then Inv^{1}(G) = 0, and if G is 1-connected then Inv^{2}(G) = 0.

Proof. From corollary 5.1.5, we have Inv^{1}(G) = H^{1}(BG, F_{p}), and of course this is Hom(π_{1}(BG), Z/p). However π_{1}(BG) = π_{0}(G). As noted above, the cycle map is injective in degree 1 over the complex numbers, and CH^{1}BG = H^{2}(BG, Z). From corollary 5.1.5 again, we see that

Inv^{2}(G) = \frac{H^{2}(BG, F_{p})}{H^{2}(BG, Z) ⊗ Z/p},

and it is elementary to show that this maps injectively onto the p-torsion in H^{3}(BG, Z) via the Bockstein. The statement about connected groups is trivial. When G is 1-connected, it is automatically 2-connected, since any real Lie group has π_{2}(G) = 0. Thus BG is 3-connected and we draw H^{3}(BG, Z) = 0 from Hurewicz’s theorem.

Example 5.2.2. Consider the case of the exceptional group G_{2}, for k = C and p = 2. It is reductive and 1-connected, so by the proposition we have Inv^{i}(G_{2}) = 0 for i = 1, 2. Moreover, the Chow ring of G_{2} has been computed over the complex numbers, see [6]. It turns out that the map CH^{2}BG_{2} ⊗_{Z} Z/2 → H^{4}(BG_{2}, F_{2}) has exactly one nonzero element. From corollary 5.1.6, we know that there is a nonzero invariant e_{3} ∈ Inv^{3}(G_{2}). It is uniquely defined as H^{3}(BG_{2}, F_{2}) = 0. It is proved in [5] that for any field k of characteristic ≠ 2, Inv(G_{2}) is in fact a free H^{∗}(k)-module on the generators 1 and e_{3}.

Example 5.2.3. We take k = C, p = 2, and G = D_{8} = Z/4 ⋊ Z/2, the dihedral group. We shall completely compute Inv(G) by showing first that things reduce to corollary 5.1.5. Let t be a generator of Z/4 in G, and let τ be the second generator, so that τ(t) = τtτ^{−1} = t^{−1}. We let G act on V = A^{2} via

t ↦ ( \begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix} )

and

τ ↦ ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

The corresponding map G → GL_{2} is an embedding, so G acts freely outside of a finite number of lines. Let U denote the open complement. We have an injection

0 → A_{G}^{0}(V, H^{∗}) = Inv(G) → A_{G}^{0}(U, H^{∗}) = A^{0}(U/G, H^{0}),

by example 2.3.2. Now, U/G is a variety of dimension 2 over an algebraically closed field, so H^{∗}(k(U)) = 0 for ∗ > 2. It follows that Inv(G) is concentrated in degrees ≤ 2. Since we are working over C, we have H_{et}^{∗}(BG) = H^{∗}(BG, F_{2}) = H^{∗}(G, F_{2}). The cohomology of G is well-known, see for example [1]:

H^{∗}(G, F_{2}) = F_{2}[x_{1}, y_{1}, w_{2}]/(x_{1}y_{1} = 0).

More precisely, G can be presented as an ”extraspecial group”, ie as a central extension

0 → Z/2 → G → E → 0

where E ≈ Z/2 × Z/2. If we choose x_{1} and y_{1} so that H^{∗}(E, F_{2}) = F_{2}[x_{1}, y_{1}], then we can pullback these classes to the cohomology of G where they will give the classes with the same name in the description above. (Besides, the cohomology class of the extention is x_{1}y_{1}.) We have immediately, by corollary 5.1.5, that Inv^{1}(G) = F_{2} · x_{1} ⊕ F_{2} · y_{1}. In degree 2, we note that H^{2}(G, F_{2}) is generated additively by the classes x_{1}^{2}, y_{1}^{2}, and w_{2}. The Chow ring of BE is CH^{∗}BE ⊗_{Z} Z/2 = F_{2}[x_{1}^{2}, y_{1}^{2}], from which we know that the classes x_{1}^{2} and y_{1}^{2} in the cohomology of BG certainly come from the Chow ring of BG. On the other hand, Sq^{1}w_{2} = w_{2}(x_{1} + y_{1}) ≠ 0 (loc. cit.), while the Steenrod operation Sq^{1} is zero on classes coming from the integral cohomology, and a fortiori it is zero on the classes coming from the Chow ring.

As a result, we have finally Inv^{2}(G) = F_{2} · w_{2}. Over a general field, we could reach a similar conclusion by studying the geometric situation a bit more carefully. The variety U/G, for example, can be shown to be the open subset in A^{2} obtained by removing the axis Y = 0 and the two parabolae X ± 2Y^{2} = 0. Alternatively, you might want to use that D_{8} is a 2-Sylow in S_{4}, and exploit the double coset formula as in [5], chap. V.

§6. Other cycle modules

We shall conclude with a few simple remarks on other possible cycle modules. Given any cycle module M as in §2, we may define the invariants Inv(G, M) of G with values in M to be the natural transformations of functors

H^{1}(−, G) → M(−).

It is straightforward to establish the inclusion

Inv(G, M) ⊂ A^{0}(BG, M)

for any BG which is the base of a versal G-principal bundle. Indeed, the arguments in [5] hold verbatim (and in fact in loc. cit. the reader will find a similar inclusion even for invariants with values in the Witt ring, even though the Witt ring satisfies weaker properties than cycle modules do.) We may also define the equivariant Chow groups A_{G}^{∗}(X, M) for varieties X acted on by G, exactly as we have done for M = H^{∗}. When this is done, we may take BG to be Spec(k)_{G} (as in the rest of this paper), and we have an equality

Inv(G, M) = A^{0}(BG, M) = A_{G}^{0}(Spec(k), M)

by Totaro’s arguments as in [5]. The techniques we have used for Galois cohomology may be used for any cycle module. Let us illustrate this for M = K_{∗}, the algebraic K-theory of fields (Milnor or Quillen, it will not affect the sequel). In this case we have A^{∗}(X, K_{0}) = CH^{∗}X, and the parallel with our previous computations becomes even more obvious. Arguing as in example 2.1.1, we obtain that A^{0}(G_{m}, K_{∗}) is a free K_{∗}-module on two generators, one in dimension 0, the other in dimension 1. If p is a prime number, we may use this to compute the invariants of μ_{p}. The group μ_{p} has a 1-dimensional representation V, and G_{m} sits in V as a μ_{p}-invariant open subset whose complement is a point Spec(k). We obtain the exact sequence (where G = μ_{p}):

0 → A_{G}^{0}(V, K_{∗}) → A_{G}^{0}(G_{m}, K_{∗}) → A_{G}^{0}(Spec(k), K_{∗−1}) → A_{G}^{1}(V, K_{∗−1}).

The action on G_{m} is free with quotient G_{m}, so as in example 2.3.2 we draw A_{G}^{0}(G_{m}, K_{∗}) = A^{0}(G_{m}, K_{∗}). Thus we may rewrite the exact sequence:

0 ⟶ Inv(G, K_{∗}) ⟶ A^{0}(G_{m}, K_{∗}) ⟶ Inv(G, K_{∗−1}),

and for ∗ = 1 it is worth writing the extra term:

0 → Inv(G, K_{1}) → A^{0}(G_{m}, K_{1}) → Inv(G, K_{0}) ⟶^{s_{∗}} CH^{1}BG → 0.

Let us explain the notation s_{∗} and what this map looks like. We call s the zero section s : BG → V_{G} of the vector bundle π : V_{G} → BG, and s_{∗} is the induced pushforward map. We know that π^{∗} is an isomorphism, and that there is a projection formula s_{∗}(π^{∗}(x)y) = xs_{∗}(y). If we use π^{∗} as an identification, it follows that s_{∗}(x) = c_{1}(V)x where c_{1}(V) = s_{∗}(1) is the first Chern class of V. Finally the map s_{∗} : Inv(G, K_{0}) → CH^{1}BG is simply the surjective map Z → CH^{1}BG sending 1 to c_{1}(V). Now, CH^{1}BG is p-torsion by a transfer argument (or see [13], example 13.1, which shows that CH^{1}BG may well be 0 depending on k). In any case, s_{∗} has a kernel isomorphic to Z. We conclude that the 1-dimensional generator for A^{0}(G_{m}, K_{∗}) must map to a generator for this kernel. As a result, Inv(G, K_{∗}) is reduced to K_{∗}, i.e. the group μ_{p} has no nonconstant invariants in algebraic K-theory at all. For p = 2 for example, assuming that char(k) ≠ 2, we may use the surjection H^{1}(K, (Z/2)^{n}) → H^{1}(K, O_{n}) for fields containing k to deduce that Inv(O_{n}, K_{∗}) injects into Inv((Z/2)^{n}, K_{∗}). Thus, after an obvious Künneth argument, we see that O_{n} has no nonconstant invariants in algebraic K-theory, either. In particular, there is no natural way of lifting the Siefel-Whitney classes to integral Milnor K-theory.

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Pierre Guillot

Université Louis Pasteur 7, rue René Descartes 67000 Strasbourg France pierre.guillot@cantab.net 546

Documenta Mathematica 12 (2007)