Steenrod operations on the Chow ring of a classifying space

Pierre Guillot

Department of Pure Mathematics and Mathematical Statistics Cambridge CB30WB

                      p.guillot@dpmms.cam.ac.uk

Abstract

We use the Steenrod algebra to study the Chow ring CH^{∗}BG of the classifying space of an algebraic group G. We describe a localization property which relates a given G to its elementary abelian subgroups, and we study a number of particular cases, namely symmetric groups and Chevalley groups. It turns out that the Chow rings of these groups are completely determined by the abelian subgroups and their fusion.

Introduction

0.1. Studying the ring CH^{∗}BG (see [32], [12]) for a complex algebraic group G, leads one to the ill-formulated conclusion that the Chow ring is the “nicest possible” subring of the cohomology – it is very often polynomial when H^{∗}BG has a lot of extra nilpotent elements. The action of the Steenrod algebra on these objects can be used to make this idea more precise.

0.2. Let 𝔘 be the category of unstable modules over A_{p}, the mod p Steenrod algebra, and let 𝔘^{ev} be the category of unstable modules which are concentrated in even dimensions. A typical object of 𝔘 is thus the mod p cohomology H^{∗}X of a topological space, while a typical object of 𝔘^{ev} is the mod p Chow ring CH^{∗}X of a smooth algebraic variety X. It is well-known that the study of 𝔘^{ev} (at any prime) is similar in many ways to that of 𝔘 at the prime 2. Indeed, the proofs of certain results in 𝔘 at p = 2 can be formally transposed to 𝔘^{ev} with practically no work, giving theorems for free. For example this is used extensively in [27]. In a sense the present paper is another illustration of this, with a connection to algebraic geometry (and Chow rings in particular) which is probably new.

Preprint submitted to Elsevier Science 20 November 2004

Since Lannes introduced the T -functor, it became clear that the cohomology ring H^{∗}ℤ/p of a cyclic group of order p played a crucial role in 𝔘. In 𝔘^{ev}, it is replaced by its even part, which is none other than CH^{∗}Bℤ/p. This is one reason why it appeared to us reasonable to suppose that Chow rings had some importance in 𝔘^{ev}. The idea was that there should be some phenomena happening specifically in 𝔘 at the prime 2 which should hold true in 𝔘^{ev} regardless of the prime if one only replaced the cohomology of elementary abelian p-groups, which are almost invariably involved in the statement of interesting results about 𝔘, by the corresponding Chow rings.

Such an example indeed presented itself, which was the starting point of this work on classifying spaces. Namely, if S_{n} denotes the symmetric group on n letters, it is established in [13] that the decomposition

H^{∗}BS_{n} = \underleftarrow{lim} H^{∗}BE

where E runs through the elementary abelian p-subgroups of S_{n}, is valid exactly at the prime 2 and not for any other p. One of our results in the present paper is that

CH^{∗}BS_{n} = \underleftarrow{lim} CH^{∗}BE

for any prime – see 2.19.

0.3. Since the existence of a decomposition as above may be elegantly regarded as a localization result in 𝔘 or 𝔘^{ev}, as we shall explain soon, the “niceness” of Chow rings alluded to in the opening paragraph can be expressed by saying that CH^{∗}BG is very often a “local” module in 𝔘^{ev} with respect to the subcategory of nilpotent modules, whereas H^{∗}BG is hardly ever local in 𝔘 (all this will be made precise in the first section).

When G is a group such that CH^{∗}BG is such a “local” module, there is surprisingly simple description of the Chow ring in terms of the cohomology ring: it will be written

CH^{∗}BG = L(ÕH^{∗}BG)

where Õ : 𝔘 → 𝔘^{ev} is right adjoint to the forgetful functor O : 𝔘^{ev} → 𝔘 and L : 𝔘^{ev} → 𝔘^{ev} is a localization functor (details in 1.10).

This is not true of all groups G: the quaternion group is a counter-example (but see 1.9 where we argue that this might be specific to the prime 2). However, we have attempted to convince the reader that good examples of groups having this property abound. In the last two sections, we prove that most Chevalley groups (definition in 3.1), including GL_{n}, Sp_{n}, SO_{n}^{+}, SL_{n} when p does not divide n, and all exceptional groups if p > 7, as well as all products and certain quotients of these, are indeed local in the above sense. Our main theorem (4.9) is the following (“𝔑𝔦𝔩-closed” is the terminology that we will use for “local”).

0.4 Theorem. Let G be a Chevalley group, let p be a prime number, and let k be a finite field of characteristic ≠ p containing the p-th roots of unity. Assume that H^{∗}(BG, ℤ) has no p-torsion. Then CH^{∗}BG(k) is 𝔑𝔦𝔩-closed in 𝔘^{ev} if G is locally isomorphic to a product of the following groups:

• Type A_{n−1}: GL_{n} for p > 2; SL_{n} if n is prime to p and p > 2;

• Type B_{n} or D_{n}: Spin_{n} for p > 2;

• Type C_{n}: Sp_{n} for p > 2;

• Exceptional type: G_{2} for p > 3; F_{4} for p > 3; E_{6} for p > 5; E_{7} for p > 7; E_{8} for p > 7.

In other words, in this situation one has

CH^{∗}BG(k) = (CH^{∗}BT (k))^{W}  (1)

where W is the Weyl group.

0.5. As indicated in the theorem, we will see that for G a Chevalley group with maximal torus T and Weyl group W, and for k a finite field, things come down to proving (1), which parallels the result for the cohomology of compact Lie groups. Here replacing the Chow rings by the cohomology of the same finite groups is certainly not possible.

That (1) is equivalent to our localization property will follow from the grouptheoretic result below, which had been obtained by Steinberg ([31]) but for which we give here a completely different, rather quick, proof which uses the computations in complex cobordism that we did in [12]:

0.6 Proposition. Let G be a Chevalley group over a finite field k of characteristic ≠ p containing the p-th roots of unity, and let T be a split maximal torus. If we assume that each elementary abelian p-subgroup of G_{ℂ} (the associated Lie group over ℂ) is contained in a maximal torus, then each elementary abelian p-subgroup of G(k) is conjugated to a subgroup of T (k).

We establish this in 3.7.

0.7. The computations that we do here complete our previous results in [12]. One general fact in this direction is that for a semi-simple Chevalley group G, the map

CH^{∗}BG(k) → MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p}

is always surjective, where MU denotes complex cobordism. (The reader should note that the precise statements in sections 3 and 4 have reasonable restrictions on p and k.) Combining this with our knowledge of the cobordism of Chevalley groups, we obtain that the image of the cycle map CH^{∗}BG(k) → H^{∗}BG(k) coincides with that of H^{∗}BG_{ℂ} → H^{∗}BG(k) for a certain “Brauer lift” BG(k) → BG_{ℂ}, where again G_{ℂ} is the associated complex Lie group over ℂ. Also, this image is a polynomial ring. See proposition 4.10.

0.8. Notations. Let us have a discussion on Chow vs cohomology rings of abelian groups; it will mostly serve as a pretext to fix the notations for the rest of the paper, but might also help the reader understand what follows.

We will always use H^{∗}X to denote the mod p cohomology of the space X; when X = BG we might occasionally write H^{∗}G. Similarly, CH^{∗}X denote the mod p Chow ring of the smooth variety X (=reduced scheme of finite type over ℂ). In [9], this would be A^{∗}X ⊗ 𝔽_{p}. See [32] for the definition of CH^{∗}BG obtained by approximating BG by smooth algebraic varieties. We may here and there use the notation CH^{∗}G. We point that, as we will see a module such as CH^{∗}X as an element in 𝔘^{ev}, we will assign to an element in CH^{k}X the degree 2k, not k.

Let p be odd. The cohomology of a cyclic group of order p is H^{∗}Bℤ/p = 𝔽_{p}[v] ⊗ Λ[u], while CH^{∗}Bℤ/p = 𝔽_{p}[v]. For an elementary abelian group we have similarly

H^{∗}(ℤ/p)^{n} = 𝔽_{p}[v_{1}, ⋯, v_{n}] ⊗ Λ(u_{1}, ⋯, u_{n})

and

CH^{∗}(ℤ/p)^{n} = 𝔽_{p}[v_{1}, ⋯, v_{n}]

Let O : 𝔘^{ev} → 𝔘 be the forgetful functor, and let Õ denote its right adjoint; if M is an unstable module, then OÕM is the largest submodule of M which is concentrated in even dimensions. For example

CH^{∗}(ℤ/p)^{n} = ÕH^{∗}(ℤ/p)^{n}  (2)

Now we make the following observation. The cohomology of a cyclic group ℤ/(p^{r}), for r ≥ 2, has the same description as in the case r = 1 above. However, as module over A_{p}, there are differences: for example, for ℤ/p we have β(u) = v, where here and elsewhere β denotes the Bockstein, whereas for ℤ/(p^{2}) we have β(u) = 0. On the other hand, the Chow rings of these groups agree as objects of 𝔘^{ev}. The relation (2) above therefore does not hold for general abelian groups. (We will prove in this paper that it does hold for S_{n}. It is rather exceptional.)

Further, the restriction map induced by the inclusion ℤ/p → ℤ/(p^{r}) is not in general an isomorphism between the cohomology rings. However this holds true for Chow rings, and we can and will replace any abelian group by its maximal elementary abelian p-subgroup when it comes to computing the mod p Chow ring.

At p = 2 now, H^{∗}ℤ/2 is a polynomial ring on one variable u in degree 1. We put v = u^{2} = β(u) = Sq^{1}(u) to have uniform notations at all primes.

0.9. Organization of the paper. In the first section we discuss localization in the categories 𝔘 and 𝔘^{ev} in some detail, in particular we explain the relation between the localization of H^{∗}BG or CH^{∗}BG and the elementary abelian subgroups of G. In section 2 we study the particular case G = S_{n}. Section 3 prepares the ground for the study of Chevalley groups, discussing Weyl groups, universal covers, and reducing the localization of these groups to the statement already mentioned. It is in section 4 that explicit computations are made with them. There is also an appendix on the double coset formula, a handy technical tool used in several places throughout the article.

1 Nilpotent modules and localization

1.1. Let G be a reductive algebraic group, and let C(G) be the category whose objects are the elementary abelian p-subgroups of G and whose morphisms are induced by conjugations in G. In [23], Quillen proved that the natural map

H^{∗}(BG) → \underleftarrow{lim}_{C(G)}H^{∗}BE

is an “F-isomorphism”, in the sense that any element in the kernel is nilpotent, and for any x in the target, x^{p^{n}} is in the image for some n. As it happens, this is a property that can be characterised in 𝔘, because the p-th power is given by a Steenrod operation. Quillen’s theorem can be reformulated by saying that the above map becomes an isomorphism upon localizing away from the subcategory of so-called nilpotent modules (we will give more precise definitions below). Rather surprisingly, it is very fruitful to “linearize” the situation in this way ([13], [15]).

Yagita proved a version of Quillen’s theorem for Chow rings when G is finite, see [33]: the map

CH^{∗}(BG) → \underleftarrow{lim}_{C(G)}CH^{∗}BE

is also an F-isomorphism. We will see in this section that we can express this as a localization result in 𝔘^{ev}, parallel to the one for cohomology. If we saw our Chow rings as modules in 𝔘 this would not be possible, which confirms our intuition that Chow rings naturally live in 𝔘^{ev}.

It is natural to ask whether there are any groups for which Quillen’s map is actually an isomorphism. For example, as already announced in the introduction, if G = S_{n} is the symmetric group, the map is an isomorphism at the prime 2 but not for any other prime, see [13]. However we shall see in the next section that the Quillen-Yagita map for S_{n} is an isomorphism at all p. Indeed the main point of this paper is to show that this is true for a whole lot of groups.

In this section we make the above statements precise, treating simultaneously the cases of 𝔘 and 𝔘^{ev}. There is essentially nothing new here, cf [15], [13], [14], but we need all this for reference. It is also felt that the reader might appreciate a concise and reasonably self-contained presentation.

1.2. Quotient categories; localizations. ([11], [10]) Let ℭ be an abelian category, and let 𝔇 be a Serre class (a full subcategory with the property that if two objects of a short exact sequence in ℭ belong to 𝔇, then so does the third). Then there is a quotient category ℭ/𝔇 which is abelian and an exact functor r : ℭ → ℭ/𝔇 satisfying the obvious universal property. A morphism in ℭ induces a monomorphism (resp. epimorphism) in ℭ/𝔇 if and only if its kernel (resp. cokernel) belongs to 𝔇. We will talk about 𝔇-monomorphisms, etc.

The category 𝔇 is said to be localizing if r admits a right adjoint, or section functor, s : ℭ/𝔇 → ℭ. We put l = s ∘ r and we call the natural map λ_{M} : M → l(M) the localization of M away from 𝔇. When this is an isomorphism, we say that M is 𝔇-closed or 𝔇-local.

One can prove easily that the natural transformation r ∘ s(M) → M is an isomorphism. It follows that λ_{M} : M → l(M) is a 𝔇-isomorphism, that is, r(λ_{M}) is an isomorphism. Hence λ_{l(M)} = l(λ_{M}) : l(M) → l ∘ l(M) is an isomorphism, too: in other words, the localization of M is local. More generally, keep in mind that a 𝔇-isomorphism M → N induces an isomorphism l(M) → l(N).

Finally, we note that if ℭ has enough injectives, then an object M is 𝔇-closed if and only if

Ext_{ℭ}^{i}(D, M) = 0

for i = 0, 1 and all D in 𝔇.

1.3. Nilpotent and 𝔑𝔦𝔩-closed modules. Given M in 𝔘 or 𝔘^{ev}, and x ∈ M of even dimension, we put P_{0}x = P^{|x|/2}x (so that if M happens to be an unstable algebra, P_{0}x = x^{p}).

We say that M is nilpotent if P_{0}^{N}x = 0 for all x of even dimension and all large N. (See [27], p47). There is an exception if we work with 𝔘 at the prime 2: in this case we put Sq_{0}x = Sq^{|x|}x and call a module M nilpotent if Sq_{0}^{N}x = 0 for all x and all large N.

The subcategory of 𝔘 or 𝔘^{ev} which is comprised of the nilpotent modules will be denoted by 𝔑𝔦𝔩 (we do not distinguish between 𝔘 and 𝔘^{ev}, hopefully the context will make things clear).

It is easy to see that 𝔑𝔦𝔩 is localizing, see the criterion in [27], prop 6.3.1. Therefore, the results of the previous paragraph apply, and we use the notation λ_{M} : M → L(M) for the localization away from 𝔑𝔦𝔩 (notation as in [15]).

It is well-known that 𝔘 has enough injectives, and it follows that the same can be said of 𝔘^{ev}. Accordingly, M is 𝔑𝔦𝔩-closed if and only if Ext^{i}(N, M) = 0 for i = 0, 1 and for all N nilpotent.

To finish with, we call a module reduced if Hom(N, M) = 0 for all nilpotent modules N. In 𝔘^{ev}, or in 𝔘 at the prime 2, this is equivalent to demanding that P_{0} be injective on M: combine lemma 2.6.4 and equation 1.7.1* in [27] (alternatively, see lemma 4.5 in [14]).

1.4 Lemma. Any reduced module in 𝔘^{ev} embeds in a reduced 𝔘^{ev}-injective. The tensor product of two reduced 𝔘^{ev}-injectives is a reduced 𝔘^{ev}-injective.

Proof. We observe immediately that if I is injective in 𝔘, then ÕI is injective in 𝔘^{ev}. The first assertion of the lemma follows then from the corresponding statement in 𝔘, which is well-known, and the left exactness of Õ. It also follows that any reduced injective in 𝔘^{ev} is a direct summand in such a module ÕI with I reduced, and we get the second assertion, again because the analogous result in 𝔘 is well-known (and because Õ(A ⊗ B) = ÕA ⊗ ÕB if one of the factors is reduced).

1.5 Proposition. In either 𝔘 or 𝔘^{ev}, we have the following properties:

(1) If 0 → M′ → M → M′′ → 0 is exact and if M′ and M′′ are both 𝔑𝔦𝔩-closed, so is M.

(2) If 0 → M → M′ → M′′ is exact and if M′ is 𝔑𝔦𝔩-closed while M′′ is reduced, then M is 𝔑𝔦𝔩-closed.

(3) M is 𝔑𝔦𝔩-closed ⟺ there exists an exact sequence

0 → M → I → J

with I and J both reduced and injective.

(4) M_{1} and M_{2} 𝔑𝔦𝔩-closed ⇒ M_{1} ⊗ M_{2} 𝔑𝔦𝔩-closed.

1.6 Corollary. Any product or inverse limit of 𝔑𝔦𝔩-closed modules is 𝔑𝔦𝔩closed.

Proof. This is trivial from the lemma above. Note that the third point in the proposition is only here to prove the fourth. In case of problems see [13] or [14].

1.7. Back to Quillen’s map. The cohomology of an elementary abelian p-group is reduced and injective ([27], 2.6.5 and 3.1.1), hence is 𝔑𝔦𝔩-closed. Consequently, the target of Quillen’s map is 𝔑𝔦𝔩-closed, as an inverse limit of such. Quillen’s theorem asserts that his map is a 𝔑𝔦𝔩-isomorphism, so that it becomes an isomorphism upon localizing. In other words

L(H^{∗}(BG)) ≈ \underleftarrow{lim}_{C(G)}H^{∗}BE

(as the inverse limit is isomorphic to its own localization). We see in this way that Quillen’s map is an isomorphism if and only if H^{∗}BG is 𝔑𝔦𝔩-closed.

Similarly, CH^{∗}BE = ÕH^{∗}BE is injective and reduced in 𝔘^{ev}, and the target of the Quillen-Yagita map is 𝔑𝔦𝔩-closed in 𝔘^{ev}. It follows that

L(CH^{∗}(BG)) ≈ \underleftarrow{lim}_{C(G)}CH^{∗}BE

and that the Quillen-Yagita map is an isomorphism if and only if CH^{∗}BG is 𝔑𝔦𝔩-closed.

1.8 Example. It is instructive to have a look at G = GL(n, ℂ), even if this G is not finite (so that the Quillen-Yagita map cannot be mentioned): in effect M = H^{∗}BG = CH^{∗}BG illustrates well how a module can behave better in 𝔘^{ev} than in 𝔘, which we see as follows. If E is the subgroup of G of diagonal matrices with p-th roots of unity as entries, then any elementary abelian subgroup of G is conjugated to a subgroup of E. If W denotes the Weyl group, it follows that

L(M) = (H^{∗}BE)^{W} ≠ H^{∗}BGL(n, ℂ);

consequently M is not 𝔑𝔦𝔩-closed in 𝔘. On the other hand

(CH^{∗}BE)^{W} = CH^{∗}BGL(n, ℂ)

from which we deduce that M is indeed 𝔑𝔦𝔩-closed in 𝔘^{ev} (being an inverse limit of 𝔑𝔦𝔩-closed modules).

To get an example with finite groups, it is in fact possible to take a finite field k of characteristic different from p but containing the p-th roots of unity, and to consider H^{∗}BGL(n, k) and CH^{∗}BGL(n, k). The former is reduced but not 𝔑𝔦𝔩-closed in 𝔘 ([15], [24]), while the latter is 𝔑𝔦𝔩-closed in 𝔘^{ev} (see [12]). In the last section of this article we shall extend this to other groups of matrices over finite fields.

1.9 Example. We cannot hope for CH^{∗}BG to be always 𝔑𝔦𝔩-closed, as is illustrated by the group of quaternions: there is only one elementary 2-subgroup, namely the centre {1; −1}, and the restriction map is neither injective nor surjective.

However, this example is not too discouraging: the quaternion groups are precisely the 2-groups which have 2-rank 1, together with cyclic groups (recall that the rank of a p-group is the dimension of the largest 𝔽_{p} vector space contained in it). For odd p on the other hand, this pathology disappears, and a group of p-rank 1 can only be cyclic. So the example above might reflect a purely group-theoretic defect (or subtlety, if you want), and one might still hope that for groups of odd order, the Quillen-Yagita map is “very often” an isomorphism.

1.10. Remark. Combining Quillen’s and Yagita’s result, we obtain of course that the cycle map CH^{∗}BG → H^{∗}BG is an F-isomorphism. So we can write in 𝔘 that L(CH^{∗}BG) = L(H^{∗}BG) and in 𝔘^{ev} that L(CH^{∗}BG) = L(ÕH^{∗}BG). Whenever CH^{∗}BG is 𝔑𝔦𝔩-closed this reads

CH^{∗}BG = L(ÕH^{∗}BG)

This is a description of the Chow ring in terms of H^{∗}BG using merely functors between 𝔘 and 𝔘^{ev} (which do not depend on G).

We end this section with a few more remarks, before starting to give examples of groups G with CH^{∗}BG 𝔑𝔦𝔩-closed, hopefully convincing the reader that there is a fair number of them. When p is odd, we do not know of an example of a group not satisfying this property.

1.11. Remark. Suppose that G is a group such that H^{∗}BG is reduced. Since the functor Õ is left-exact and commutes with inverse limits, we have an injection:

ÕH^{∗}BG ↪ \underleftarrow{lim} ÕH^{∗}BE

But ÕH^{∗}BE = CH^{∗}BE, so if we suppose further that the Quillen-Yagita map for Chow rings is surjective, it follows that the image of CH^{∗}BG under the cycle map has to contain all of ÕH^{∗}BG. We will use this later with G = S_{n}.

1.12. Complex cobordism. Recall ([32]) that the cycle map factors as

CH^{∗}BG → MU^{∗}(BG) ˆ⊗_{MU^{∗}}𝔽_{p} → H^{∗}BG

where MU denotes complex cobordism. Most of what we have said so far applies to the theory MU^{∗}(−) ˆ⊗_{MU^{∗}}𝔽_{p}.

We start by explaining why we have Steenrod operations on this theory. First, we have the isomorphism of graded abelian groups:

MU^{∗}(MU) = MU^{∗}(pt)[[⋯ c_{n} ⋯]]

More generally for any oriented cohomology theory E, the Thom-Dold theorem asserts E^{∗}MU = E^{∗}BU, and E^{∗}BU is computed via the Atiyah-Hirzebruch spectral sequence, and is a power series ring as above. However this is not a ring isomorphism: we are interested in the multiplication given by composition in MU^{∗}(MU) = [MU, Σ^{∗}MU]. Nevertheless, it is a fact (cf [26],VII,3.1) that any element x ∈ MU^{∗}(MU) can be written

x = ∑_{ω} a_{ω}S_{ω}

where a_{ω} ∈ MU^{∗}(pt) and S_{ω} has an (easy) explicit description in terms of the c_{i}’s. The ring structure on MU^{∗}(pt) is the natural one, and the notation a_{ω}S_{ω} does refer to the “correct” multiplicative structure. From this it follows that the kernel I of the (surjective) map MU^{∗}(MU) → H^{∗}MU, where as always H denotes mod p cohomology, is the ideal generated by (p, x_{i}) ⊂ MU^{∗}(pt), in standard notation. In other words, I is generated in MU^{∗}(MU) by the kernel of the map

MU^{∗}(pt) = ℤ[x_{1}, x_{2}, ⋯] → 𝔽_{p} = H^{∗}(pt)

Now, this I acts trivially on MU^{∗}(X) ˆ⊗_{MU^{∗}}𝔽_{p} for any space X, of course. (That is, the elements of this ideal act as 0.) Thus H^{∗}MU acts on our theory (again, with some ring structure which is not the one coming from the isomorphism H^{∗}MU = H^{∗}BU, but this will not matter). Since the Steenrod algebra acts on H^{∗}MU, it also acts on MU^{∗}(X) ˆ⊗_{MU^{∗}}𝔽_{p}. Moreover, it is well-known that H^{∗}MU is a free A_{p}/(β)-module, so that the Bocksteins act trivially in this new setting. (Note that H^{∗}MU is thus not unstable, this is an example that shows that the Thom-Dold isomorphism is not a map of A_{p}-modules!)

Is MU^{∗}(X) ˆ⊗_{MU^{∗}}𝔽_{p} unstable? This appears to be true, and we outline a proof – we shall not need the result in what follows, as we shall only consider some spaces X for which MU^{∗}(X) ˆ⊗_{MU^{∗}}𝔽_{p} injects into the cohomology ring. To prove the claim, one would construct directly some operations on MU^{∗}(−) ˆ⊗_{MU^{∗}}𝔽_{p} in the obvious way: starting with a manifold M representing a class in MU^{∗}X, then M^{p} gives a class in MU^{∗}X^{p}, and thus it yields one in MU^{∗}Z^{p}X, the cyclic product; project this onto

MU^{∗}(X × Bℤ/p) ˆ⊗_{MU^{∗}}𝔽_{p} = (MU^{∗}(X) ˆ⊗_{MU^{∗}}𝔽_{p}) ⊗ (MU^{∗}(Bℤ/p) ˆ⊗_{MU^{∗}}𝔽_{p})

and define the coefficients of the polynomial in v thus obtained to be your “Steenrod” operations. These act “unstably” for any space, by construction. After taking limits, we extend this to X = MU, and the images of 1 ∈ MU^{∗}(MU) ˆ⊗_{MU^{∗}}𝔽_{p} define elements P̄^{i} there; these lift to MU^{∗}(MU) and yield operations on MU^{∗}(−) and on MU^{∗}(−) ˆ⊗_{MU^{∗}}𝔽_{p} which by naturality agree with the ones just described. Thanks to the existence of such lifts, in fact, any element in MU^{∗}(MU) ˆ⊗_{MU^{∗}}𝔽_{p} acts on MU^{∗}(−) ˆ⊗_{MU^{∗}}𝔽_{p}, and thus any relation between the P̄^{i}’s and the c_{i}’s still holds after it’s applied to any x in MU^{∗}(X) ˆ⊗_{MU^{∗}}𝔽_{p} for any X. By picking some spaces X for which MU^{∗}(X) ˆ⊗_{MU^{∗}}𝔽_{p} injects into the cohomology H^{∗}X, and using the obvious compatibility of the P̄^{i}’s with the actual Steenrod operations P^{i} once we have projected to cohomology, we deduce P̄^{i} = P^{i}.

In any case, if we take this for granted, we see that the modules MU^{∗}(X) ˆ⊗_{MU^{∗}}𝔽_{p} live naturally in the category 𝔘/β of unstable modules over A_{p}/(β), which is an intermediate between 𝔘^{ev} and 𝔘. Considering the odd and even parts of a module M in 𝔘/β gives a decomposition of M as a direct sum of two elements of 𝔘^{ev}. It should be easy from this to extend the localization results obtained for 𝔘 and 𝔘^{ev} to the category 𝔘/β. However, we do not have any application in sight, mostly because all modules of interest to us given as MU^{∗}(BG) ˆ⊗_{MU^{∗}}𝔽_{p} for some G are concentrated in even dimensions and can be seen as objects of 𝔘^{ev}. Therefore we shall not fill in the details here. The interested reader might want to have a look at [22], formula 3.18 in particular.

In any case, it is convenient to use the term “F-isomorphism”, and we state for future reference Yagita’s result just alluded to:

1.13 Proposition. Let G be a finite group. The natural map

MU^{∗}(BG) ˆ⊗_{MU^{∗}}𝔽_{p} → \underleftarrow{lim} MU^{∗}(BE) ˆ⊗_{MU^{∗}}𝔽_{p}

is an F-isomorphism.

1.14 Corollary. If G is a finite group, then the two maps

CH^{∗}BG → MU^{∗}(BG) ˆ⊗_{MU^{∗}}𝔽_{p} → H^{∗}BG

are F-isomorphisms.

2 The Symmetric Groups

2.1. In this section we prove that CH^{∗}BS_{n} is 𝔑𝔦𝔩-closed in 𝔘^{ev}. As one could expect, the proof is by induction, by proving that if CH^{∗}BG is 𝔑𝔦𝔩-closed, then so is CH^{∗}B(S_{p} ≀ G). It might seem simpler to use ℤ/p ≀ G instead, but we want the reader to be able to compare our proof with that in [14] which deals with a similar result using ÕH^{∗}BG instead of our CH^{∗}BG (and in turn, both treatments follow closely the original one in [13] for mod 2 cohomology). Incidentally, the result in [14] can be recovered from ours, see 2.20.

2.2. Assumptions. For technical reasons stemming from [32], we will have to restrict attention (until section 3) to a certain class of groups: namely, G will be assumed to have a subgroup H of index prime to p such that BH can be cut into open subsets of affine space. Then ℤ/p ≀ H is a subgroup of index prime to p in S_{p} ≀ G which has the same property, by [32], lemma 8.1, and the induction may proceed.

We will also assume that the mod p cycle map CH^{∗}BG → H^{∗}BG is injective. We will prove quickly that the cycle map for ℤ/p ≀ G and S_{p} ≀ G is injective too.

For instance, note that these two assumptions are satisfied in the case of abelian groups. In fact, all that matters to prove the result on S_{n} is that they hold for the trivial group.

2.3. We begin by explaining how Chow rings and cohomology rings are affected by taking wreath products, and we give a few immediate properties, in particular we will end up with an exact sequence which together with 1.5,1/, will eventually yield the result.

2.4. Cohomology and Chow rings of cyclic products. We shall need the following nice result of Nakaoka[20]:

H^{∗}(ℤ/p ≀ G) = H^{∗}(ℤ/p, (H^{∗}G)^{⊗p})

Recall that the cohomology of a cyclic group with coefficients in any ring A is periodic, and the period is given by taking cup-products with an element in H^{2}(ℤ/p, A). Here we denote by v ∈ H^{2}(ℤ/p, (H^{∗}G)^{⊗p}) an element giving the period (abusing the notation given in the introduction).

For Chow rings of cyclic products, we use results of [32]. There a certain functor F_{p} from graded abelian groups to graded abelian groups is defined, which comes equipped with a natural map F_{p}CH^{∗}X → CH^{∗}Z^{p}X, where X is a variety and Z^{p}X is it p-fold cyclic product. After changing the grading from dimension to codimension, reducing modulo p, and changing the notations to relate to [5], the definition of F_{p}A^{∗} is as follows: take the p-fold tensor product A^{∗} ⊗ ⋯ ⊗ A^{∗}, take symbols P x in degree p · |x| for x ∈ A^{∗} of positive degree, take also elements α_{i}x of degree p · |x| + 2i for all x ∈ A^{∗} and positive i, and finally divide by the relations:

x_{1} ⊗ ⋯ ⊗ x_{p} = x_{2} ⊗ ⋯ ⊗ x_{p} ⊗ x_{1}
x^{⊗p} = 0
P (x + y) = P x + P y + ∑ s_{1} ⊗ ⋯ ⊗ s_{p}

together with the relations that turn α_{i} into a homomorphism of groups. Here the sum in the third formula runs over a set of representatives for the ℤ/porbits in the set {x, y}^{p}−{(x, ⋯, x), (y, ⋯, y)}. To be rigorous, the elements α_{i}x should be only defined for i small enough and likewise the P x’s should only appear when the degree of x is small enough; but in both cases the bound depends on the dimension of the variety X, and when we deal with classifying spaces of groups BG we take a limit of varieties (with “compatible” Chow rings) having their dimensions going to infinity, and we do not need to worry about this complication.

From the construction of F_{p} as given in [32], it is easy to describe the composition F_{p}CH^{∗}BG → CH^{∗}B(ℤ/p ≀ G) → H^{∗}B(ℤ/p ≀ G), for any G: the element x_{1} ⊗ ⋯ ⊗ x_{p} is sent to the “norm” ∑ x_{i_{1}} ⊗ ⋯ ⊗ x_{i_{p}}, with the sum running over all cyclic permutations, sitting in H^{0}(ℤ/p, (H^{∗}G)^{⊗p}); the element P x is sent to x ⊗ ⋯ ⊗ x in the same group (note how the relations above relate to the expansion of (x + y) ⊗ ⋯ ⊗ (x + y)); and finally the element α_{i}x goes to v^{i} · x ⊗ ⋯ ⊗ x. This is all clear. In particular this map is injective if the cycle map for G is injective.

For G as in the introduction, lemma 8.1 in [32] implies that F_{p}CH^{∗}BG → CH^{∗}B(ℤ/p ≀ G) is surjective. Therefore for the G we consider, there is an isomorphism F_{p}CH^{∗}BG = CH^{∗}B(ℤ/p ≀ G), the cycle map for ℤ/p ≀ G is injective, and its image is explicitly described.

We shall denote by τ the transfer from G^{p} to ℤ/p ≀ G. Its image is spanned by the “norms”, clearly.

2.5. We will also be interested in wreath product of the form S_{p} ≀ G. Nakaoka has established that H^{∗}(S_{p} ≀G) = H^{∗}(S_{p}, (H^{∗}G)^{⊗p}) in this case too. Since ℤ/p is a p-Sylow of S_{p}, we deduce:

2.6 Lemma. Let W = N_{S_{p}}(ℤ/p)/ℤ/p = (ℤ/p)^{∗}. Then

H^{∗}(S_{p} ≀ G) = (H^{∗}(ℤ/p ≀ G))^{W}

and

CH^{∗}(S_{p} ≀ G) = (CH^{∗}(ℤ/p ≀ G))^{W}

Proof. The equality for cohomology groups follows from Nakaoka’s results just quoted and Swan’s lemma. To get the result for Chow rings, we proceed as follows: there is an obvious inclusion, and the double coset formula (see the appendix) tells us that the image of the restriction map is the group of “stable” elements, ie those x such that, putting K = ℤ/p ≀ G, we have x|_{K∩gKg^{−1}} = gxg^{−1}|_{K∩gKg^{−1}} (proposition A.5). So we need to show that if x is W -invariant, then x is stable. But K ∩gKg^{−1} is either K or G^{p} (this is because G^{p} is normal in S_{p} ≀ G, and the order of ℤ/p is prime). In either case, the cycle map is injective on this group, and so the result for cohomology implies that for Chow rings.

2.7 Proposition. There are exact sequences:

0 → τ(H^{∗}(BG^{p})) → H^{∗}B(ℤ/p ≀ G) → H^{∗}BG ⊗ H^{∗}Bℤ/p

and

0 → τ(CH^{∗}(BG^{p})) → CH^{∗}B(ℤ/p ≀ G) → CH^{∗}BG ⊗ CH^{∗}Bℤ/p

Note that the maps on the right come from the inclusion of ℤ/p×G in ℤ/p≀G = ℤ/p ⋉ G^{p}.

Proof. The exact sequence for cohomology follows from results of Steenrod’s ([30] chapter VII), as is explained, for example, in [19], theorem II.3.7.

To get the result for Chow rings, we use the cycle map, which here is injective. The only thing to prove is that CH^{∗}B(ℤ/p≀G)∩τ(H^{∗}(BG^{p})) = τ(CH^{∗}(BG^{p})), but this is clear from the explicit description in 2.4.

2.8 Corollary. There are exact sequences:

0 → (τ(H^{∗}BG^{p}))^{W} → H^{∗}B(S_{p} ≀ G) → R_{1}(H^{∗}BG) → 0

and

0 → (τ(CH^{∗}BG^{p}))^{W} → CH^{∗}B(S_{p} ≀ G) → R_{1}^{ev}(CH^{∗}BG) → 0

where W is as in lemma 2.6 and where R_{1}(H^{∗}BG), resp. R_{1}^{ev}(CH^{∗}BG), is a submodule of H^{∗}BG ⊗ H^{∗}Bℤ/p, resp. CH^{∗}BG ⊗ CH^{∗}Bℤ/p.

The functorial notations R_{1}(H^{∗}BG) and R_{1}^{ev}(CH^{∗}BG) will be justified below.

2.9. The point now is to prove that (τ(CH^{∗}BG^{p}))^{W} and R_{1}^{ev}(CH^{∗}BG) are both 𝔑𝔦𝔩-closed, and to use proposition 1.5, 1/. This is where we can only continue the proof for Chow rings, as the result does not hold for cohomology at odd primes. It is proved in [14], however, that if one replaces H^{∗}BK by ÕH^{∗}BK for all groups K occuring in Quillen’s map, then one still gets an isomorphism.

2.10. The functor R_{1}^{ev}. Let M be a module in 𝔘^{ev}. For any x ∈ M of degree 2k, define

St_{1}^{ev}(x) = ∑_{i=0}^{k}(−1)^{i}v^{i(p−1)} ⊗ P^{k−i}x ∈ CH^{∗}Bℤ/p ⊗ M

where we recall that P^{i} = Sq^{2i} when p = 2. We define R_{1}^{ev}M to be the CH^{∗}(BS_{p})-submodule of CH^{∗}Bℤ/p⊗M generated by the elements St_{1}^{ev}x, for x ∈ M. Here we view CH^{∗}Bℤ/p as a module over CH^{∗}BS_{p} via the restriction map.

There is a functor R_{1} : 𝔘 → 𝔘 which is defined in a similar, but more complicated, way (see for example [14], definition 4.2, or the original in [34]). In fact one has R_{1}^{ev}M = ÕR_{1}(OM): this can be seen from the explicit description of R_{1} given in loc cit , and using lemma 2.6 which asserts in particular that CH^{∗}S_{p} = 𝔽_{p}[v^{p−1}]. This proves that R_{1}^{ev}M is always in 𝔘^{ev}, ie it is stable under the action of the Steenrod algebra. We will not use this seriously, however, and R_{1}^{ev} can be taken as a functor from 𝔘^{ev} to graded 𝔽_{p}-vector spaces.

The definition of R_{1}^{ev} is very explicit and will allow computation. However:

2.11 Lemma. The two definitions of R_{1}^{ev}M given coincide.

Proof. We need to show that the image of the map

CH^{∗}(S_{p} ≀ G) → CH^{∗}Bℤ/p ⊗ CH^{∗}G

is R_{1}^{ev}M. The fact that the elements St_{1}^{ev}x are in this image follows from the very definition of the Steenrod operations on Chow rings: in [5], an element P (x) is constructed in CH^{∗}(S_{p} ≀G) (prop 4.2 in loc cit) which retricts to St_{1}^{ev}x (definition 7.5 in loc cit; note that our (−1)^{i} sign is on p10, before prop 6.6 there). Therefore, the image certainly contains R_{1}^{ev}M.

To get the reverse inclusion, observe that (with notations as in 2.4, our P x being consistent with Brosnan’s) the elements in CH^{∗}(ℤ/p ≀ G) not mapping to 0 in CH^{∗}G ⊗ CH^{∗}Bℤ/p are of the form P x or α_{i}x = v^{i} · P x. The latter elements can only be in CH^{∗}(S_{p} ≀ G) = (CH^{∗}(ℤ/p ≀ G))^{W} (cf 2.6) if i is a multiple of (p − 1), and the image of the map above is indeed contained in the CH^{∗}BS_{p} = F_{p}[v^{p−1}]-module generated by the St_{1}^{ev}x.

2.12. A few properties of R_{1}^{ev}. Put P = CH^{∗}Bℤ/p and Q = CH^{∗}BS_{p}.

2.13 Lemma. If {x} is an 𝔽_{p}-basis of M, then {St_{1}^{ev}x} is a basis of R_{1}^{ev}M as a free Q-module.

Proof. (Almost word for word from [17], proof of 4.2.3, included for the convenience of the reader). Given a relation ∑ λ_{x}St_{1}^{ev}x = 0 with λ_{x} ∈ P, put m = min |x| taken over all x for which λ_{x} ≠ 0, if there are any. Then project into P ⊗ M^{m}, which is a free P -module having a basis containing the elements 1 ⊗ x for x of degree m. You get, with m = 2k, the relation ∑ λ_{x}(−1)^{k}v^{k(p−1)} ⊗ x = 0 which yields λ_{x} = 0.

2.14 Lemma. The functor R_{1}^{ev} is exact.

Proof. From the previous lemma, we can say that as a vector space, R_{1}^{ev}M is Q ⊗ ΦM, where here ΦM means M with all degrees mutiplied by p. Result follows.

2.15 Lemma. IF M′ is a submodule of M, then (R_{1}^{ev}M)∩(P ⊗M′) = R_{1}^{ev}M′.

Proof. This follows from the commutative diagram with exact rows:

0 ⟶ R_{1}^{ev}M′ ⟶ R_{1}^{ev}M ⟶ R_{1}^{ev}M/M′ ⟶ 0
↓  ↓  ↓
0 ⟶ P ⊗ M′ ⟶ P ⊗ M ⟶ P ⊗ M/M′ ⟶ 0

Our next lemma will involve the functor Φ defined in [27], 1.7. When M ∈ 𝔘^{ev}, ΦM is M with all degrees multiplied by p, and with an appropriate action of the Steenrod algebra which makes the map ΦM → M, x ↦ P_{0}x linear over A_{p} (so more formally, P^{i}Φx = ΦP^{i/p}x if p|i and 0 otherwise). Whenever M is reduced, we identify ΦM with the submodule of M comprised of the elements P_{0}x.

2.16 Lemma. For any M ∈ 𝔘^{ev}, we have

(R_{1}^{ev}ΦM) ∩ (ΦP ⊗ M) ⊂ ΦR_{1}^{ev}M

Proof. (cf [14], 4.7). Let x ∈ M have degree 2k; we can write

St_{1}^{ev}Φx = ∑_{j=0}^{k}(−1)^{j}v^{jp(p−1)} ⊗ P^{(k−j)p}Φx

because P^{kp−i} can only act non-trivially on Φx when p divides i (so we have put i = pj). A typical element in (R_{1}^{ev}ΦM) ∩ (ΦP ⊗ M) is thus a sum of elements of the form

y = v^{pm(p−1)} ∑_{j=0}^{k}(−1)^{j}v^{jp(p−1)} ⊗ P^{(k−j)p}Φx

and this is Φz for

z = v^{m(m−1)} ∑_{j=0}^{k}(−1)^{j}v^{j(p−1)} ⊗ P^{(k−j)}Φx

from Cartan’s formula and 1.7.1* in [27].

2.17 Proposition. If M ∈ 𝔘^{ev} is 𝔑𝔦𝔩-closed, so is R_{1}^{ev}M.

Proof. We prove that if an element y of R_{1}^{ev}M is of the form y = P_{o}x for some x ∈ CH^{∗}Bℤ/p ⊗ M, then in fact we can choose such an x in R_{1}^{ev}M. It follows that P_{0} is injective on the quotient (CH^{∗}Bℤ/p ⊗ M)/R_{1}^{ev}M, so that this module is reduced (cf 1.3). As CH^{∗}Bℤ/p ⊗ M is 𝔑𝔦𝔩-closed from proposition 1.5, 4/, it follows from 2/ of the same propostion that R_{1}^{ev}M is 𝔑𝔦𝔩-closed.

We have

(R_{1}^{ev}ΦM) ∩ (ΦP ⊗ M) ⊂ ΦR_{1}^{ev}M

and

R_{1}^{ev}M ∩ (P ⊗ ΦM) = R_{1}^{ev}ΦM

from 2.15 & 2.16. Thus

R_{1}^{ev}M ∩ Φ(P ⊗ M) ⊂ ΦR_{1}^{ev}M

using the fact that on 𝔘^{ev}, Φ commutes with tensor products, just like it does on 𝔘 at the prime 2, so that Φ(P ⊗ M) = (ΦP ⊗ M) ∩ (P ⊗ ΦM).

This was what we wanted.

2.18. By contrast, proving that τ(CH^{∗}BG^{⊗p})^{W} is 𝔑𝔦𝔩-closed is straightforward. Put M = CH^{∗}BG. The explicit formulae of 2.4 and 2.6 show that there is an exact sequence

0 → τ(M^{⊗p}) → (M^{⊗p})^{ℤ/p} → ΦM → 0

Since M is reduced, direct computation shows that ΦM is reduced; the middle term of the sequence is 𝔑𝔦𝔩-closed by 1.5,4/ and 1.6; so τ(M^{⊗p}) is 𝔑𝔦𝔩-closed by 1.5, 2/; and finally (τ(M^{⊗p}))^{W} is 𝔑𝔦𝔩-closed by 1.6 again.

2.19 Theorem. Let G be a group as in the introduction. If CH^{∗}BG is 𝔑𝔦𝔩closed, so is CH^{∗}B(S_{p} ≀ G). In particular, CH^{∗}BS_{n} is 𝔑𝔦𝔩-closed for any n. It follows that

CH^{∗}BS_{n} = \underleftarrow{lim} CH^{∗}BE

where E runs over the elementary abelian subgroups of S_{n}.

Proof. The only thing to add is that the Sylow subgroup of S_{n} is contained in a product of iterated wreath products S_{p} ≀ S_{p} ≀ ⋯ ≀ S_{p}.

2.20. It has been known since Quillen’s paper [23] that the cohomology of S_{n} is reduced. From 1.11, we deduce

CH^{∗}BS_{n} = ÕH^{∗}BS_{n}

It follows that

ÕH^{∗}BG = \underleftarrow{lim} ÕH^{∗}BE

which was established in [14], a paper which has been a great source of inspiration.

3 Localization of Chevalley groups

3.1. Chevalley groups. The point of this section and the next is to study Chevalley groups, ie groups of the form G(𝔽_{l^{a}}) where G is a connected, reductive, split group scheme over ℤ, and l is a prime different from p. Recall that “split” means that G has a maximal torus T which is itself “split” in the sense that T = 𝔾_{m}^{n} over ℤ (or whatever the base is); in this definition 𝔾_{m} is the multiplicative group scheme Spec ℤ[X, X^{−1}].

3.2. The Weyl group. We have to start with some general remarks on group schemes. Recall that normalisers, centralisers, and quotients (at least by diagonalisable subgroups) can be performed over any base scheme, see [1], exposé VIII,6/. It will be important to us that these operations commute with base extensions – for example if G is a group scheme over S and H a subgroup scheme, and if R → S is any morphism, then (N_{G}(H))_{R} = N_{G_{R}}(H_{R}) where X_{R} = X ×_{S} R.

If now G is a reductive group scheme over S with a split maximal torus T, recall that T = C(T) (its own centraliser), cf [2], exposé XIX, 2.8. We (momentarily) put W_{G}(T) = N_{G}(T)/T. The following is proved in [2], exposé XXII,3/ (in particular proposition 3.4): there exists a finite group W such that W_{G}(T) is the constant group scheme associated to W. This means in particular that for any ring R above S without idempotents other than 0 and 1, we have W_{G}(T)(R) = W. Furthermore, it is also established in loc cit that

W ⊂ \frac{N_{G}(T)(S)}{T (S)} ⊂ W_{G}(T)(S)

Using the preceding remark on base extensions, we conclude that the above inclusions are equalities whenever S is replaced by (the spectrum of) a ring without nontrivial idempotents.

Now let G be a Chevalley group as in the previous paragraph. There is thus a finite group W such that

\begin{aligned}W & = & \frac{N_{G}(T)(ℤ)}{T (ℤ)} & = & \frac{N_{G}(T)}{T}(ℤ) \\ & = & \frac{N_{G}(T)(k)}{T (k)} & = & \frac{N_{G}(T)}{T}(k) \end{aligned}

for any field k.

It follows that W acts on T (k). Note that unless k is algebraically closed, there is no reason for N_{G}(T)(k) to be the normaliser of T (k) in G(k), it may be a strictly smaller subgroup; similarly T (k) = C(T)(k) may be smaller than the centraliser of T (k) in G(k). However we have the following

3.3 Lemma. Any automorphism of T (k) induced by conjugation by an element of G(k) may also be realised by an element of W. More generally, any isomorphism between two subgroups of T (k) induced by conjugation by an element of G(k) can also be realised by conjugation by an element of W.

Proof. This is a well-known argument. So let K denote an algebraic closure of k. Suppose x ∈ G(k) and xAx^{−1} = B. Let C be the connected component of 1 in the centraliser of B in G(K). We have then T (K) and xT (K)x^{−1} as maximal tori in C, and therefore they are conjugated by some c ∈ C ⊂ G(K). Consider then n = c^{−1}x. It is clear that n ∈ N_{G}(T)(K), and can be written n = wt with w ∈ N_{G}(T)(k) and t ∈ T (K), according to the equalities above, valid for any field. Thus the elements w and n induce the same automorphism of T (K), and induce the same isomorphism between A and B as x does.

3.4. Assumptions & Notations. Throughout the rest of the paper, G will be a Chevalley group, T a split maximal torus, N_{T} will be short for N_{G}(T), W will be the finite group just introduced (to be refered to as the Weyl group), and finally k will be a finite field of characteristic l different from p but containing the p-th roots of unity. The associated Lie group over ℂ will be denoted G_{ℂ}, and it is assumed that H^{∗}(BG_{ℂ}, ℤ) has no p-torsion (see below). Some of this will be repeated for emphasis.

3.5. Localization of Chevalley groups. The elementary abelian subgroups of Lie groups have been studied extensively by Borel [3], among others. He proves the following amazing theorem:

3.6 Theorem. Let Γ be a compact connected Lie group. Then the following three conditions are equivalent:

(1) H^{∗}(Γ, ℤ) has no p-torsion,

(2) H^{∗}(BΓ, ℤ) has no p-torsion,

(3) any elementary abelian subgroup of Γ is contained in a maximal torus.

The reader might find it amusing to see how the implication 2/ ⇒ 3/ can be proved very quickly using some recent results in homotopy theory: namely, if E is elementary abelian, then conjugacy classes of maps E → Γ are precisely the same as maps H^{∗}BΓ → H^{∗}BE of unstable algebras ([16], corollaire 3.1.4); the assumption on p implies, classically, that H^{∗}BΓ → H^{∗}BT is injective, where T is any maximal torus, and thus what we need to prove is that we can extend the map H^{∗}BΓ → H^{∗}BE induced by the inclusion of E into Γ to a map H^{∗}BT → H^{∗}BE of unstable algebras. In turn, this follows from the “non-linear injectivity of H^{∗}BE”, [27], 3.8.7.

We prove now a variation on Borel’s theorem, which will explain why it is convenient to restrict attention to those primes p such that H^{∗}(BG_{ℂ}, ℤ) has no p-torsion. We point out that, if K is a maximal compact subgroup of G_{ℂ}, then K and G_{ℂ} are homotopy equivalent, as are BK and BG_{ℂ}; therefore the above theorem shows that our assumption is equivalent to H^{∗}(G_{ℂ}, ℤ) having no p-torsion.

3.7 Proposition. Let G be a Chevalley group with split maximal torus T, let p be a prime number such that H^{∗}(BG_{ℂ}, ℤ) has no p-torsion, and let k be a finite field of characteristic ≠ p which contains the p-th roots of unity. Then any elementary abelian p-subgroup of G(k) is conjugated to a subgroup of T (k).

Proof. It is proved in [12] that for such k, we have

MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p} = H^{∗}BG_{ℂ}

for any G (hence for T as well). By choice of p, the map H^{∗}BG_{ℂ} → H^{∗}BT_{ℂ} is injective, so that

MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p} → MU^{∗}(BT (k)) ˆ⊗_{MU^{∗}}𝔽_{p}

is injective. Combining this with proposition 1.13 tells us that the natural map

\underleftarrow{lim}_{C(G)}MU^{∗}(BE) ˆ⊗_{MU^{∗}}𝔽_{p} → \underleftarrow{lim}_{C′(G)}MU^{∗}(BE) ˆ⊗_{MU^{∗}}𝔽_{p}

is an F-monomorphism, where C′(G) is the subcategory of C(G) (definition in section 1) consisting of those E which are conjugated to a subgroup of T (k). (In fact this is simply a monomorphism, as both its source and target are 𝔑𝔦𝔩-closed modules in 𝔘^{ev}.)

Suppose that there is a V ∈ C(G) which is not in C′(G). Choose a V maximal with respect to this property, and note that V is then maximal in C(G). We construct an element x in MU^{∗}(BV) ˆ⊗_{MU^{∗}}𝔽_{p} which restricts to 0 in any proper subgroup of V: for this, choose for each such subgroup E a non-zero element x_{E} in MU^{∗}(BV) ˆ⊗_{MU^{∗}}𝔽_{p} which restricts to 0 in MU^{∗}(BE) ˆ⊗_{MU^{∗}}𝔽_{p}, for example the first Chern class of a non-trivial 1-dimensional representation of V which is trivial on E; and then take x to be the product of all the different x_{E}’s. By symmetry, x is invariant under the action of the normaliser of V. (In fact, in this way we end up taking x to be the product of all non-zero elements of degree 2, but we prefer to phrase it this way.)

Given this, define α(E) = 0 if E ∈ C(G) is not conjugated to V and α(E) = x otherwise, with an obvious abuse of notation. By maximality of V and choice of x, this defines an element α in the inverse limit on the left hand side above (we need such an x because there could be a group W which is not conjugated to V but having a subgroup E conjugated to a subgroup of V).

This is a contradiction, as α is non-zero (ie non-nilpotent) but it lies in the kernel of the (F-) monomorphism above. Hence C(G) = C′(G).

3.8 Corollary. The localization of CH^{∗}BG(k) away from 𝔑𝔦𝔩 is

L(CH^{∗}BG(k)) = (CH^{∗}BT (k))^{W}

Remark (1). Here’s a variant of the proposition. Keep the same hypotheses but do not assume that k contains the p-th roots of unity; then each elementary abelian p-subgroup of G(k) is contained in a maximal torus, possibly not split. To see this, let E be such a subgroup, and let K be an algebraic closure of k. Let Z(E, G) be the centraliser of E in G(K), and let Z^{0} be the connected component of 1 – a reductive group. From the result obtained when k is big enough, we deduce that E is contained in Z^{0}. Being central, it is contained in any maximal torus of Z^{0}, and there is one which is defined over k.

Remark (2). Our result can be recovered from a theorem of Steinberg, as follows. Let E be an elementary p-subgroup of G(k) and let K be an algebraic closure of k. Then theorem 2.28 in [31] says that E is toral over K, ie there is a g ∈ G(K) such that gEg^{−1} ⊂ T (K) where T is the fixed, split maximal torus. Considering the assumption on k, gEg^{−1} is in fact contained in T (k). Now consider the set X of x ∈ G(K) such that xex^{−1} = geg^{−1} for all e in E. It is a principal homogeneous space under Z, the centraliser of E in G(K). This Z is defined over k and connected (theorem 2.28 in loc cit again), and by Lang, it has a k-rational point. In other words there is an x ∈ G(k) such that xEx^{−1} is contained in T (k), which was what we wanted.

Note. The two remarks above are due to J.P. Serre.

3.9. Definition. We will call condition (1), resp. (2), the injectivity, resp. surjectivity, of

CH^{∗}BG(k) → (CH^{∗}BT (k))^{W}

We will see shortly that (1) or (2) holds for G if it holds for its universal cover G̃, with a partial converse.

It might be enlightening, to start with, to recall what is known about the cobordism of Chevalley groups.

3.10. Cobordism. The cobordism of Chevalley groups is completely described in [12]; in particular, the result already quoted says

MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p} = H^{∗}BG_{ℂ}

(a natural isomorphism with respect to maps of group schemes over ℤ.) It is also established that the latter rings inject into H^{∗}BG(k), and are polynomial.

We deduce first of all that MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p} can be viewed as an element in 𝔘^{ev}. If p is odd, we can say a bit more: a theorem of Dwyer-Miller-Wilkerson ([7], 2.11) implies then that

MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p} = (MU^{∗}(BT (k)) ˆ⊗_{MU^{∗}}𝔽_{p})^{W}

(in fact equality holds if and only if the left hand side is polynomial, cf [18], lemma 7.1.) In other words, MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p} is 𝔑𝔦𝔩-closed in 𝔘^{ev}.

3.11. Covers. Assume that G is semi-simple and consider now the universal cover G̃ of G, ie the unique Chevalley group such that G̃_{ℂ} is the universal cover of G_{ℂ}. The centre of a semi-simple Chevalley group is finite and contained in any maximal torus, hence is a product of groups μ_{a} (a-th roots of unity). These groups have the peculiarity that all subgroups of μ_{a}(ℂ) are defined over ℤ, and the same can be said thus of the subgroups of the centre Z(ℂ) of G̃_{ℂ}. In particular π_{1}(G_{ℂ}) can be seen as a finite and central subgroup π of G̃, defined over ℤ. Now, the groups G̃/π and G agree over ℂ, and hence agree over ℤ, by the structure theorem for Chevalley groups, cf [2], exposé XXV, théorème 1.1.

The map G̃(k) → G(k), where as always k is finite, may not be as nice as one would expect it to be (eg it is almost never surjective). Its kernel, at least, is certainly a subgroup of π(k), so it is central. The order of μ_{a}(k) divides that of μ_{a}(ℂ), and it follows that |π(k)| divides |π(ℂ)| = |π_{1}(G_{ℂ})| (this is a result that holds for more general groups, but it is trivial to check it directly here). Now let T ors(G) denote the set of prime numbers l such that H^{∗}(BG_{ℂ}, ℤ) has l-torsion. All prime numbers dividing the order of π_{1}(G_{ℂ}) are in T ors(G), in fact T ors(G) = T ors(G̃) ∪ {l : l divides |π_{1}(G_{ℂ})|}: see [28], 1.3.1 to 1.3.4, and the references there, in particular [3]. So our standing assumption that H^{∗}(BG_{ℂ}, ℤ) have no p-torsion implies that the order of π(k), and thus that of the kernel of G̃(k) → G(k), is prime to p.

We also have to note that |G̃(k)| = |G(k)|: this is very specific to finite fields, and is proved in [4], proposition 16.8. It follows that the image of G̃(k) in G(k) has index prime to p.

We note:

3.12 Lemma. Let Γ be given as a quotient Γ̃/C where C is a central subgroup of the finite group Γ̃. Assume that the order of C is prime to p. Then H^{∗}Γ = H^{∗}Γ̃ and CH^{∗}BΓ = CH^{∗}BΓ̃.

Proof. A spectral sequence argument gives the result immediately for cohomology. The result for Chow rings follows from remark A.6, but of course it is preferable to find an elementary argument (even if we only use the easier part of A.6). We let S̃ and S denote Sylow subgroups of Γ̃ and Γ respectively. Clearly S̃ maps isomorphically to (a conjugate of) S under the quotient map; we now use proposition A.5. To prove that the stable subgring for S is sent isomorphically onto the stable subring for S̃ under the isomorphism CH^{∗}BS → CH^{∗}BS̃, things come down to checking the following: if x̃ ∈ S̃ and if g̃ ∈ G̃(k), then g̃x̃g̃^{−1} is in S̃ if and only if gxg^{−1} is in S, where x and g correspond to x̃ and g̃ under the projection Γ̃ → Γ. To see the non-trivial part of this, assume that gxg^{−1} ∈ S and write g̃x̃g̃^{−1} = c · s with c ∈ π(k) and s ∈ S̃. Writing o(σ) for the order of an element σ, we have o(c · s) = o(c)o(s) because c and s commute and have coprime orders. But o(c · s) is a power of p, as this element is conjugate to x̃. Therefore o(c) = 1 and c = 1.

Applying this with Γ taken to be the image of G̃(k) in G(k) shows that CH^{∗}BG(k) is a direct summand in CH^{∗}BG̃(k) (in 𝔘^{ev}). So if CH^{∗}BG̃(k) is reduced or 𝔑𝔦𝔩-closed, so is CH^{∗}BG(k). In fact, we can say something more precise.

Letting Γ be as above, we claim that G(k) is the product of T (k) and Γ (although these subgroups have a non-trivial intersection, of course). To see this, let K be an algebraic closure of k; we may assume to have chosen a maximal torus T̃ in G̃ such that p(T̃(K)) = T (K), where p : G̃ → G is the natural map. Since T̃(k) contains the kernel C of the map G̃(k) → G(k), it follows easily that the image E of T̃(k) in T (k) is all of Γ ∩ T (k). Moreover, since T̃(k) and T (k) have the same order (by the result already quoted, or directly), it follows that [T (k) : E] = [G(k) : Γ] (in turn these indices equal the order of C). Consequently, the number of products γ · t with γ ∈ Γ and t ∈ T (k) is

\frac{|Γ| · |T (k)|}{|Γ ∩ T (k)|} = |Γ| · [T (k) : E] = |Γ| · [G(k) : Γ] = |G(k)|

Thus G(k) = Γ · T (k), as claimed. We apply now the double coset formula (see the appendix), which gives here:

i_{T (k)→G(k)}^{∗} ∘ i_{∗}^{Γ→G(k)} = i_{∗}^{E→T (k)} ∘ i_{E→Γ}^{∗}

Since E has index prime to p in the abelian group T (k), it is clear that i_{∗}^{E→T (k)} is an isomorphism. The transfer i_{∗}^{Γ→G(k)} is surjective for index reasons. Thus we see from the formula above that the surjectivity of CH^{∗}BG(k) → (CH^{∗}BT (k))^{W} is equivalent to that of CH^{∗}BΓ → (CH^{∗}BE)^{W}. From the last lemma applied to Γ and E, this is also equivalent to G̃ satisfying (2). Summarizing:

3.13 Proposition. Let G̃ be the universal cover of the semi-simple Chevalley group G. If G̃ satisfies (1), resp. (2), then G satisfies (1), resp. (2). Moreover if G satisfies (2), then so does G̃.

Remark. It is clear from the discussion above that the proposition holds a bit more generally: if G̃ → G is an isogeny (over ℤ) whose kernel C is central and has |C(k)| prime to p, then the conclusion of the proposition holds.

Remark (on notation). The simply-connected group Spin_{n} has a central subgroup μ_{2}, and the quotient Spin_{n}/μ_{2} is the orthogonal group SO_{n}. Note that the group we call SO_{n}(k) is usually denoted SO_{n}^{+}(k), while SO_{2n}(ℝ) sometimes goes under the name SO(n, n) (and SO_{2n+1}(ℝ) can be SO(n + 1, n)). The familiar group of orthogonal motions of ℝ^{n} with determinant 1, normally denoted SO(n), is the compact form of the group we consider.

4 Chevalley groups: examples

4.1. When p does not divide |W |. We assume that G is semi-simple and as usual that the characteristic of k is not p. Under these hypotheses, a theorem of Springer-Steinberg ([29] 5.19) asserts that a p-Sylow S of G(k) normalises a maximal torus T′ defined over k. This T′ does not have to be split; however if we suppose further that p does not divide the order of the Weyl group of G, it follows that S is abelian. If we denote its normaliser in G(k) by N_{S}, we have by Swan’s lemma (A.7)

CH^{∗}BG(k) = (CH^{∗}BS)^{N_{S}}

Now, let S′ be the maximal p-elementary subgroup of S. We have (cf 0.8) CH^{∗}BS = CH^{∗}BS′. Let N_{S′} denote the normaliser of S′ in G(k). Clearly N_{S} ⊂ N_{S′} (since S′ is precisely the subgroup of S of elements of order p, it is preserved by any automorphism of S) and thus (CH^{∗}BS′)^{N_{S′}} ⊂ (CH^{∗}BS′)^{N_{S}}. However since there are isomorphisms

CH^{∗}BG(k) → (CH^{∗}BS)^{N_{S}} → (CH^{∗}BS′)^{N_{S}}

we obtain the reverse inclusion, and CH^{∗}BG(k) = (CH^{∗}BS′)^{N_{S′}}.

Now, S′ being a p-elementary subgroup of S of maximal rank, it is also of maximal rank in G(k); since (by 3.7) we know that S′ is (conjugated to) a subgroup of T (k) where T is our fixed, split maximal torus, we deduce that S′ is precisely the maximal elementary abelian subgroup of T (k) and that CH^{∗}BT (k) = CH^{∗}BS′ (0.8 again).

Finally, the group of automorphisms of CH^{∗}BS′ induced by N_{S′} is the same as that induced by N_{T} (k) (or W), by lemma 3.3. Thus:

4.2 Proposition. Suppose G is semi-simple. If p does not divide |W |, then CH^{∗}BG(k) is 𝔑𝔦𝔩-closed, ie

CH^{∗}BG(k) = (CH^{∗}BT (k))^{W}

Example. Consider the five exceptional Lie groups G_{2}, F_{4}, E_{6}, E_{7} and E_{8}. Their Weyl groups have orders 12, 2^{7}.3^{2}, 2^{7}.3^{4}.5, 2^{10}.3^{4}.5.7 and 2^{14}.3^{5}.5^{2}.7 respectively. The primes which give torsion in their integral cohomology are {2} for G_{2}, {2, 3} for F_{4}, E_{6} and E_{7}, and {2, 3, 5} for E_{8} (see [28]). In other words, we can say that CH^{∗}BG(k) is always 𝔑𝔦𝔩-closed when G is exceptional and does not have p-torsion in its integral cohomology, except possibly in finitely many cases, namely when (G, p) is either (G_{2}, 3), (E_{6}, 5), (E_{7}, 5), (E_{7}, 7), or (E_{8}, 7). Restricting attention to p > 7 rules out these cases.

4.3. Chern classes. Another case of interest is that of a group G such that H^{∗}BG_{ℂ} is generated by Chern classes of (finitely many) representations of G_{ℂ}. In this case we have:

4.4 Proposition. Suppose that H^{∗}BG_{ℂ} is generated by Chern classes, and let p be odd. Then the map

CH^{∗}BG(k) → MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p}

is surjective, as is

CH^{∗}BG(k) → (CH^{∗}BT (k))^{W}

Proof. By assumption there is a surjective map

⨂_{i} H^{∗}BGL(n_{i}, ℂ) → H^{∗}BG_{ℂ}

Such representations of G_{ℂ} must be defined over ℤ (essentially because irreducible representations of G over ℤ are classified by their highest weights, just like those of G_{ℂ} over ℂ; and as G is split, the weights are the same over ℤ or ℂ. Alternatively, appeal to [2], XXV, 1.1.). Consider the following commutative diagram:

⨂ CH^{∗}BGL(n_{i}, k)  ⟶  CH^{∗}BG(k)  ⟶^{c}  (CH^{∗}BT (k))^{W}
a↓^{≈}  _{b}↓  ↓^{≈}
⨂ MU^{∗}(BGL(n_{i}, k)) ˆ⊗_{MU∗} 𝔽_{p} ⟶^{d} MU^{∗}(BG(k)) ˆ⊗_{MU∗} 𝔽_{p} ⟶^{e} (MU^{∗}(BT (k)) ˆ⊗_{MU∗} 𝔽_{p})^{W}

The map a is an isomorphism by the computation of CH^{∗}BGL(n, k) that was made in [12]. Moreover d is surjective by the remark just made and the results of loc cit. It follows that b is surjective. Finally, e is an isomorphism by (3.10), so c is surjective.

Example. Groups such that H^{∗}BG_{ℂ} is generated by Chern classes include SL_{n}, Sp_{n}, and SO_{2n+1} for odd p’s. Here again we point out that SO_{2n+1} has 2-torsion in its integral cohomology, so the restriction that p be odd in the above proposition is not important anyway; however for SL_{n} and Sp_{n}, we are leaving out cases where the conclusion of the proposition might still be true.

4.5. The orthogonal groups. We have already established that, with some restrictions on p, all simple, simply connected groups satisfy condition (2), with the notable exception of Spin_{2n}. Replacing this group by SO_{2n} as we may, we can use an ad hoc argument. Recall that we are interested in odd primes since the spin/orthogonal groups have 2-torsion in their cohomology.

We refer back to [12] where it is proved that a certain continuous map BSO_{2n}(k) → BSO_{2n}ℂ induces isomorphisms

MU^{∗}(BSO_{2n}ℂ) ˆ⊗_{MU^{∗}}𝔽_{p} ⟶^{≈} MU^{∗}(BSO_{2n}(k)) ˆ⊗_{MU^{∗}}𝔽_{p} ⟶^{≈} (MU^{∗}(BT (k)) ˆ⊗_{MU^{∗}}𝔽_{p})^{W}

Each of these rings is a polynomial ring over 𝔽_{p} on variables c_{2i} (1 ≤ i ≤ n−1) and e. By definition then, these classes are the Pontryagin classes and the Euler class of the oriented bundle E → BSO_{2n}(k) induced by the classifying map BSO_{2n}(k) → BSO_{2n}ℂ. Now if this bundle were algebraic, it would have Pontryagin and Euler classes in CH^{∗}BSO_{2n}(k) which would restrict to the elements with the same name in CH^{∗}BT (k), proving that (2) holds (recall that CH^{∗}BT (k) = MU^{∗}(BT (k)) ˆ⊗_{MU^{∗}}𝔽_{p}, of course, and that the cobordim modules are 𝔑𝔦𝔩-closed, cf 3.10). However E need not be algebraic.

The Pontryagin classes are Chern classes and one can argue as in 4.3 to prove that they exist in CH^{∗}BSO_{2n}(k). We need more work for the Euler class.

So let S be a p-Sylow of SO_{2n}(k). It is proved in [6] that any map BP → BG, where p is a finite p-group and G a compact Lie group, comes from a homomorphism P → G, up to homotopy. Therefore the composition BS → BSO_{2n}(k) → BSO_{2n}ℂ is homotopic to a map coming from a homomorphism S → SO_{2n}ℂ. As S is finite, this is automatically a map of algebraic groups, and the map BS → BSO_{2n}ℂ is algebraic. Using this map to pull back the universal vector bundle over BSO_{2n}ℂ, we get an algebraic vector bundle over BS carrying a quadratic form and which is equivalent to E (or rather, to the pull back of E over BS). By [8], this bundle has an Euler class in the Chow ring, ie there is a class in CH^{∗}BS mapping under the cycle map to the Euler class in MU^{∗}(BS) ˆ⊗_{MU^{∗}}𝔽_{p} (up to scalar multiplication by a power of 2, but we are still assuming that p is odd). Call this x ∈ CH^{∗}BS, put e′ = i_{∗}^{S→SO_{2n}(k)}(x), and finally let e = i_{T (k)→SO_{2n}(k)}^{∗}(e′). To prove that this is the Euler class defined above (and justify the notation), it is enough to check this in MU^{∗}(BT (k)) ˆ⊗_{MU^{∗}}𝔽_{p}, that is, after applying the cycle map. But in cobordism we have Euler classes for topological bundles, and x = i_{S→SO_{2n}(k)}^{∗}(y) for some y ∈ MU^{∗}(BSO_{2n}(k)) ˆ⊗_{MU^{∗}}𝔽_{p}, so that e′ = y up to a nonzero scalar (we abuse the notations x and e′ here). It follows that e is indeed the Euler class that we want.

4.6. Condition (1) and wreath products. The Weyl groups of the simple, simply-connected Lie groups have a lot in common. Namely, for the classical groups at least, W always contains a copy of the symmetric group acting on the torus by permuting the eigenvalues. This often gives N_{T} (k) a tractable group structure. The following lemma is trivial, but it is remarkable to note how often it applies.

4.7 Lemma. Suppose that G(k) contains a subgroup of index prime to p which is of the form S_{r} ⋉T (k) = S_{r} ≀k^{∗} (here r is the rank of G). Then the restriction map:

CH^{∗}BG(k) → CH^{∗}BT (k)

is injective.

Proof. (Compare with 2.19.) The restriction to S_{r} ≀ k^{∗} is injective for index reasons. If follows from [12], lemma 4.4, that S_{r} ≀ k^{∗} possesses a collection of abelian p-groups which detect the Chow ring – in the sense that if x ∈ CH^{∗}B(S_{r} ≀ k^{∗}) restricts to 0 in each of these subgroups then x = 0. Thus the Chow ring is also detected, in the same sense, by the family of elementary abelian p-subgroups. As observed above, these are conjugated to subgroups of T (k).

Remark. The reader will have noticed that an easy induction based on 2.19 will suffice to prove that S_{r} ≀ k^{∗} has a reduced Chow ring. However, the proof in [12] and a number of results in the same paper yield the slightly stronger statement that this group possesses a subgroup H of index prime to p whose Chow ring is reduced and which satisfies a certain condition (called (*) there) meaning in fact the hypotheses of 2.2 together with the property that the integral cycle map to cohomology is injective. In turn, two such groups H_{1} and H_{2} have a Künneth formula, in the sense that

CH^{∗}BH_{1} × BH_{2} = CH^{∗}BH_{1} ⊗ CH^{∗}BH_{2}

which is not true for the Chow ring of a product of varieties in general, but see loc cit and [32] in this case.

It follows that if the lemma applies for two Chevalley groups G_{1} and G_{2}, then their product also has a reduced Chow ring.

Example (1). It is easily seen that the lemma applies, for odd p, for the groups Sp_{n} and Spin_{n}.

Example (2). In some cases we can also deal with SL_{n}. Here W = S_{n} acts in the following way: if we see T (k) as {(x_{1}, ⋯, x_{n}) ∈ (k^{∗})^{n} : x_{1} ⋯ x_{n} = 1} (that is, as a subgroup of the maximal torus for GL_{n}), then S_{n} acts by permutation of the x_{i}’s. Here N_{T} (k) is not a wreath product, and not even a semidirect product. However, if we look at the subgroup of W which fixes the last coordinate, we see that it is isomorphic to S_{n−1} and that its preimage in N_{T} (k) is N′ = S_{n−1} ≀ k^{∗} = T (k) ⋊ S_{n−1}. Indeed, GL_{n−1}(k) contains a copy of S_{n−1} acting on the diagonal matrices by permuting the eigenvalues, and which consists of matrices of determinant ±1; and so we find, in the evident way, a copy of S_{n−1} in SL_{n}(k) acting on diagonal matrices by permuting their first n − 1 entries. This gives the required splitting of N′.

This subgroup has index n in N_{T} (k), which in turn has index prime to p in SL_{n}(k) when p is odd. We conclude that the lemma applies for SL_{n}(k) when p is odd and prime to n.

4.8. Summary. Let us put together the information gathered so far. First of all, there is the

4.9 Theorem. Let G be a Chevalley group, let p be a prime number, and let k be a finite field of characteristic ≠ p containing the p-th roots of unity. Assume that H^{∗}(BG, ℤ) has no p-torsion. Then CH^{∗}BG(k) is 𝔑𝔦𝔩-closed in 𝔘^{ev} if G is locally isomorphic to a product of the following groups:

• Type A_{n−1}: GL_{n} for p > 2; SL_{n} if n is prime to p and p > 2;

• Type B_{n} or D_{n}: Spin_{n} for p > 2;

• Type C_{n}: Sp_{n} for p > 2;

• Exceptional type: G_{2} for p > 3; F_{4} for p > 3; E_{6} for p > 5; E_{7} for p > 7; E_{8} for p > 7.

In other words, in this situation one has

CH^{∗}BG(k) = (CH^{∗}BT (k))^{W}

where W is the Weyl group.

Here “locally isomorphic” refers to the situation in 3.11 (see the remark following the proposition there: this is why it makes sense to include GL_{n}, which is not simply-connected, in the list).

Our main omission with simply-connected groups is SL_{n} when p divides n. If one could prove the result in this case as well, we would be able to say that CH^{∗}BG(k) is 𝔑𝔦𝔩-closed for any semi-simple group (and all large p).

Nevertheless, we have the following proposition. We formulate it for p > 7 in order to obtain a concise statement, but the reader who prefers an exhaustive list of results as above will easily bring the necessary modifications himself.

4.10 Proposition. Let p > 7, let G be any semi-simple Chevalley group whose integral cohomology has no p-torsion, and let k be any finite field of characteristic ≠ p. Let K be the finite field obtained from k by adjoining the p-roots of unity and let r = [K : k].

Then the map

CH^{∗}BG(k) → MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p}

is surjective. Moreover for a certain Brauer lift BG(k) → BG_{ℂ}, the image of H^{∗}BG_{ℂ} → H^{∗}BG(k) coincides with the image of the Chow ring under the cycle map. Writing H^{∗}BG_{ℂ} = 𝔽_{p}[s_{1}, ⋯, s_{n}], this image is

𝔽_{p}[s_{i} : 2r divides |s_{i}|].

Proof. The point is that we have established (2) for any simply-connected group and for p > 7, hence for any semi-simple group by 3.11. The first statement follows immediately from this and 3.10, at least for K; but the map MU^{∗}(BG(K)) ˆ⊗_{MU^{∗}}𝔽_{p} → MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p} is always surjective as we showed in [12].

Everything else in the proposition was proved in loc cit for cobordism, with the exception of the injectivity of

MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p} → H^{∗}BG(k)

which we need to describe the image and which is only stated there for K. The more general statement is obvious though, since MU^{∗}(BG(k)) ˆ⊗_{MU^{∗}}𝔽_{p} is a polynomial ring for any k and since the cycle map to cohomology in an F -monomorphism.

Acknowledgements. This work will be part of my PhD thesis and I would like to thank Burt Totaro for his help as my supervisor. I am also indebted to J-P. Serre for pointing out alternative approaches to the study of elementary subgroups of Chevalley groups (see the two remarks in 3.7).

A Appendix: the double coset formula

A.1. The double coset formula is a useful equality that shows up whenever a functor h^{∗}(−) from groups to rings presents itself equipped with transfers. Of course, examples for h^{∗}(−) include CH^{∗}B−, MU^{∗}(B−) ˆ⊗_{MU^{∗}}𝔽_{p} and H^{∗}(−); example for transfers include not only the standard one, but also Even’s norm used by Quillen in [25] and by Yagita in [33].

The lemma that follows implies the double coset formula (to be stated shortly) in all cases of interest. We submit a proof because it is perhaps not wellknown that it can be stated in the category of algebraic varieties (as opposed to topological spaces), which is what we need when working with Totaro’s definition of a classifying space ([32]). In turn, this will fill a very minor gap in Yagita’s proof alluded to in section 1, which implicitly uses the double coset formula for Chow rings and for the Even’s multiplicative norm.

A.2. Let G be a finite group, and let K, H be subgroups. Choose representative elements σ_{i} of the K-H-double cosets, that is, such that G is the disjoint union of the sets Kσ_{i}H. The intersection K ∩ σ_{i}Hσ_{i}^{−1} will be written L_{i}.

We let U be a (Zariski) open set in a vector space, on which G acts freely. For each subgroup A of G, the variety U/A is Totaro’s approximation to BA, and this is what the notation will mean in what follows.

A.3 Lemma. The following square is a pull-back:

∐_{i} BL_{i} ⟶ BH
↓  ↓
BK ⟶ BG

Proof. All maps should have an obvious definition. Let X = {(x, y) ∈ BK × BH : p_{1}(x) = p_{2}(y)} where p_{1} and p_{2} are the obvious maps to BK and BH respectively. Of course X = BK ×_{BG} BH. We first define a map BL_{i} = U/L_{i} ⟶ X ⊂ U/K ×U/H by ū ↦ (ū, \overline{σ_{i}^{−1} · u}). This is well-defined and takes values in X; moreover one checks easily that it gives a bijection ∐_{i} BL_{i} → X. As we work over a field of characteristic 0, and as all varieties encountered are normal (in fact, smooth – for example X is étale over BK or BH), this is an isomorphism of algebraic varieties.

A.4. The formula As we have just pointed out, there are several versions of the double coset formula, and generalisations to any algebraic group over any field, etc. In all cases it reads:

i_{K→G}^{∗} ∘ i_{∗}^{H→G} = ∑_{i} i_{∗}^{L_{i}→K} ∘ i_{L_{i}→σ_{i}Hσ_{i}^{−1}}^{∗} ∘ c_{σ_{i}^{−1}}

One deduces it easily from the last lemma (cf prop 1.7 in [9] for Chow rings).

We now list a few consequences. Proofs are standard and will be omitted.

A.5 Proposition. Let G be a finite group, let G_{p} be a subgroup of index prime to p. Then CH^{k}(G) restricts isomorphically onto the “stable” subgroup of CH^{k}(G_{p}). In other words, defining a map CH^{k}(G_{p}) → CH^{k}(G_{p} ∩ G_{p}^{g}) by

Ψ_{g}(x) = i_{G_{p}∩G_{p}^{g}→G_{p}}^{∗}(x) − i_{G_{p}∩G_{p}^{g}→G_{p}^{g}}^{∗} ∘ c_{g^{−1}}^{∗}(x)

then the following sequence is exact:

0 → CH^{k}(G) → CH^{k}(G_{p}) → ∏_{g∈G} CH^{k}(G_{p} ∩ G_{p}^{g})

A.6 Remark. Let G and G′ be finite groups, and let S and S′ be Sylows. The Martino-Priddy conjecture, recently proved by Bob Oliver ([21]), asserts that whenever there is a homotopy equivalence BG_{p}^{∧} ≈ BG′_{p}^{∧}, then there is an isomorphism of groups S ≈ S′ which is “fusion preserving”, ie it respects the conjugacies in the strongest possible sense. It follows then from the last proposition that there is an isomorphism CH^{∗}BG ≈ CH^{∗}BG′. Note that any map G → G′ inducing an isomophism on cohomology yields a homotopy equivalence between the p-completed classifying spaces, and thus any homomorphism of groups which induces an isomorphism on cohomology also implies the existence of an isomorphism between the Chow rings (or cobordism rings, or any ring coming from a theory with transfers for that matter).

However we shall refrain from using this result, as the proof of the Martino-Priddy “conjecture” is very long and complicated and proceeds by a case-by-case check through the classification of finite simple groups.

A.7 Proposition (Swan’s lemma). Suppose that the p-Sylow subgroup G_{p} is abelian. Then if N_{p} is its normalizer, we have

CH^{∗}(G) = CH^{∗}(G_{p})^{N_{p}}

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