16.3.1993

LOCALIZATIONS OF UNSTABLE A - MODULES AND EQUIVARIANT MOD p COHOMOLOGY

by

Hans–Werner Henn, Jean Lannes and Lionel Schwartz

Introduction

Let p be a fixed prime and G be either a compact Lie - group (not necessarily connected) or a discrete group of finite virtual cohomological dimension (f.v.c.d. for short), e.g. the general linear group over the ring of S-integers in a number field or the mapping class group of an orientable surface. In [Q1] Quillen considered the category A(G) whose objects are the elementary abelian p - subgroups of G and whose morphisms are given as compositions of inclusions and conjugations. He showed that the restriction homomorphisms from the mod p cohomology H^{∗}BG of the classifying space BG to H^{∗}BE for E ∈ A(G) induce a map q_{G} : H^{∗}BG ⟶ lim_{A(G)^{op}} H^{∗}BE such that the kernel of q_{G} consists of nilpotent elements and such that for each element in the limit a sufficiently large p^{n} - th power is in the image of q_{G}. (As usual, A(G)^{op} denotes the opposite category of A(G), so that E ↦ H^{∗}BE is a covariant functor on A(G)^{op}.) In this paper we will consider H^{∗}BG as an unstable module over the mod p Steenrod algebra A (unstable module for short) and show how this structure can be used to refine Quillen’s result and describe a finite sequence of approximations to H^{∗}BG starting with Quillen’s map q_{G} and ending with a genuine isomorphism. To do this we consider the full subcategory Nil_{n} of the category U of all unstable modules; Nil_{n} is the smallest subcategory of U which contains all n - fold suspensions and is closed with respect to forming extensions and taking filtered colimits. Here is an explanation for our terminology. The unstable module underlying an unstable algebra (e.g. the mod p - cohomology of a space) is an object in Nil_{1} if and only if all its elements are nilpotent in the usual sense. The subcategory Nil_{n} is localizing, i.e. there exist a localization functor L_{n} and a natural transformation λ_{n} : id_{U} ⟶ L_{n}, the localization away from Nil_{n}. Quillen’s map q_{G} is actually localization away from Nil_{1}. The sequence of approximations referred to above will be the sequence of localizations away from Nil_{n} and the following result shows that this sequence of localizations is in many cases actually finite.

THEOREM 0.1 [He]. Let K be an unstable algebra which is finitely generated as an algebra. Then the localization away from Nil_{n} is an isomorphism for all sufficiently large n.

The localization L_{n}H^{∗}BG and the localization map λ_{n} can be roughly described as follows. As noted above the map λ_{1} arose from the product of restriction homomorphisms H^{∗}BG ⟶ ∏_{E∈A(G)} H^{∗}BE, and L_{1}H^{∗}BG was given by the subalgebra of ∏_{E∈A(G)} H^{∗}BE consisting of those families of elements {x_{E}}_{E∈A(G)} which satisfy the compatibility conditions imposed by the morphisms in Quillen’s category A(G). The description of λ_{n} is similar. Let C_{G}(E) denote the centralizer of E in G and (H^{∗}BC_{G}(E))^{<n} denote the unstable algebra which is obtained from H^{∗}BC_{G}(E) by dividing out the ideal of all elements of degree at least n. Then λ_{n} arises from the map λ̃_{n} : H^{∗}BG ⟶ ∏_{E∈A(G)} H^{∗}BE ⊗ (H^{∗}BC_{G}(E))^{<n} induced by the group homomorphisms E × C_{G}(E) → G, (e, g) ↦ e · g. (Observe that λ̃_{1} identifies with the previous map H^{∗}BG ⟶ ∏_{E∈A(G)} H^{∗}BE.) Here is a preliminary version of our description of L_{n}H^{∗}BG and of λ_{n}.

THEOREM 0.2. Let G be a compact Lie group or a discrete group of f.v.c.d. such that H^{∗}BG is a noetherian algebra.

a) The kernel of the map

λ̃_{n} : H^{∗}BG ⟶ ∏_{E∈A(G)} H^{∗}BE ⊗ (H^{∗}BC_{G}(E))^{<n}

is the largest A - invariant submodule of H^{∗}BG which is in Nil_{n}.

b) The elements in the image of λ̃_{n} satisfy suitable compatibility conditions defining a subalgebra Λ_{n} of ∏_{E∈A(G)} H^{∗}BE ⊗(H^{∗}BC_{G}(E))^{<n}; the map λ_{n} : H^{∗}BG ⟶ L_{n}H^{∗}BG identifies with the map H^{∗}BG ⟶ Λ_{n} induced by λ̃_{n}.

The precise compatibility conditions are somewhat complicated and we refer the reader to Theorem I.5.4 below for a rigorous statement of part (b) of Theorem 0.2. Roughly speaking, Theorem 0.2 says that L_{n}H^{∗}BG is determined by A(G) and the mod p cohomology of the spaces BC_{G}(E) for all E ∈ A(G) in degrees less than n. The following immediate consequence of Theorem 0.1 and Theorem 0.2 deserves to be emphasized.

COROLLARY 0.3. Let G be as in Theorem 0.2. For n sufficiently large the map

λ̃_{n} : H^{∗}BG ⟶ ∏_{E∈A(G)} H^{∗}BE ⊗ (H^{∗}BC_{G}(E))^{<n}

is a monomorphism.

For further applications to the study of H^{∗}BG the reader is referred to part I, section 5.

These results suggest to introduce the following two invariants of an unstable A - module M. Let

d_{0}M := inf{n ∈ |λ_{n+1}M is a monomorphism}

and

d_{1}M := inf{n ∈ |λ_{n+1}M is an isomorphism} .

The following result gives a crude but very easy to state general estimate of these invariants in case M = H^{∗}BG. For refinements we refer the reader to sections II.2 and II.3.

THEOREM 0.4. Let G be a compact Lie group and let U be a unitary group such that G embedds into U. Then we have the following inequalities

a) d_{0}H^{∗}BG ≤ dim U − dim G

b) d_{1}H^{∗}BG ≤ 2 dim U − dim G .

The paper consists of two parts. Part I deals with localizations of unstable modules over the mod p - Steenrod algebra. In section 1 and 2 we introduce notation and identify the quotient categories U/Nil_{n} with certain categories of analytic functors defined on the category E of finite dimensional vector spaces over the field _{p} with p elements. This generalizes the results of [HLS1, HLS2] where the category U/Nil_{1} was identified with the category of analytic functors from E to the category of all _{p} vector spaces. In section 3 we give a description of the localization functors L_{n} in terms of the functors T_{V} introduced by the second author in [L1, L2]. Furthermore we introduce the invariants d_{0} and d_{1} and the tower of localization functors. The description of L_{n}M given in section 3 can be substantially simplified if the unstable module M satisfies suitable finiteness assumptions, for example if M is a noetherian unstable algebra. These simplifications are carried out in section 4. Up to this point our theory is completely algebraic. In section 5 we use as first geometric input the computation of T_{V} of the mod p cohomology H_{G}^{∗}X := H^{∗}(EG ×_{G} X; _{p}) [L3]; here G is a compact Lie group, X a finite G - CW - complex and EG ×_{G} X the usual Borel construction. This input allows us to translate the results of the previous sections, in particular of section 4, and to derive the precise version of Theorem 0.2 (cf. I.5.4) in the more general situation of equivariant cohomology. As an example of Theorem 0.2 (resp. I.5.4) we show how the general description of L_{n}H^{∗}BG simplifies in case each element of order p in G is central (Corollary I.5.8). Furthermore we indicate how our theory can be used to get structural information on H^{∗}BG. The reader mainly interested in cohomology of groups is advised to begin his reading with this section.

In part II we are concerned with the study of the invariants d_{0} and d_{1} in the case of equivariant cohomology as well as with discussing explicit examples. In section 1 we show how a result of Duflot [D1] can be used to prove the inequality d_{1}H_{V}^{∗} M ≤ dimM if M is a smooth compact manifold equipped with a smooth action of an elementary abelian p - group V. This is then used in section 2 to give estimates for d_{0} and d_{1} in case of H_{G}^{∗}M where M is a compact manifold with a smooth action of a compact Lie group G. In fact, compactness can be weakened in both cases but smoothness is important. In any case we prove refined versions of Theorem 0.4 in this section. Section 3 discusses the symmetric construction in case p = 2 in which case the estimates available from section 2 can often be significantly sharpened. In section 4 we consider explicit examples and give the values of d_{0}H^{∗}BG and d_{1}H^{∗}BG for various groups like finite abelian, dihedral, quaternion and semidihedral groups. Finally in section 5 we illustrate our theory in the case of the cohomology of classifying spaces of various general linear groups. In particular, we show how our theory can be used to recover Quillen’s computation of H^{∗}(BGL(n, _{q}); _{2}), the mod 2 cohomology of the general linear groups over a finite field _{q} with q elements, if q ≡ 3 mod 4.

The authors are grateful to the following institutions for support and for enabling the authors to work together on this project: Université Paris XI, Institut d’Estudis Catalans and Universität Heidelberg. In addition the first author would like to thank the IHES and the Heisenberg program of the DFG for support.

I. Localizations of unstable modules

1. The categories U/Nil_{n}, F^{<n} and the functors f^{<n} and m^{<n}

Let p be a fixed prime and denote by A the mod p Steenrod algebra. We begin by recalling some basic definitions.

\underline{1.1.} A graded A - module M is called unstable if the following holds: if p = 2 then Sq^{i}x = 0 for all i > |x|; if p is odd and e ∈ {0, 1} then β^{e}P^{i}x = 0 for all 2i + e > |x|. Here |x| denotes the degree of the element x. The category of unstable modules and A - linear maps will be denoted by U.

\underline{1.2.} If M is an A-module, then its suspension ΣM is the A-module whose underlying graded vector space is defined by (ΣM)^{n} = M^{n−1} and whose A-module structure is given by θΣx = (−1)^{|x|}Σθx, θ ∈ A. (As usual x ∈ M^{n−1} is denoted Σx if considered as element in (ΣM)^{n}.) If M is unstable, then ΣM is also unstable, but, of course, the converse fails, and this failure is the point of departure for this paper.

DEFINITION 1.3. An unstable A-module N is called n-nilpotent iff every finitely generated submodule admits a finite filtration such that each filtration quotient is an n-fold suspension.

\underline{1.4.} We note that 1 - nilpotent is nilpotent in the sense of [HLS1,HLS2]. Furthermore n - nilpotent in the sense of this definition is (n − 1) - nilpotent in the sense of [S1]. There is also a characterization of n - nilpotent modules in terms of Steenrod operations as follows [S1,S2]: If p = 2 the operations

Sq_{k} : N^{m} → N^{2m−k}, x ↦ Sq^{m−k}x

have to be locally nilpotent (i.e. act nilpotently on each element x) for all 0 ≤ k < n; if p is odd we write k = 2l + e with e ∈ {0, 1} and replace Sq_{k} by

P_{l} : N^{2m+e} → N^{2mp+e−2l(p−1)}, x ↦ P^{m−l}x .

If n = 1 and N is the mod-p-cohomology of a space this means that the p - th power map is locally nilpotent in the classical sense. This fact justifies the terminology.

\underline{1.5.} The full subcategory of U whose objects are the n - nilpotent modules will be denoted by Nil_{n}. Clearly this is the smallest “Serre class” in U which contains all n - fold suspensions and is closed with respect to taking colimits. For our purposes it is very important that these subcategories can be described in terms of certain injective objects in U as follows.

If V is an elementary abelian p-group, we will often write H^{∗}V for the mod p -cohomology of its classifying space BV. Furthermore we have the Brown-Gitler modules J(k) which can be characterized as representing objects for the functors M ↦ (M^{k})^{∗} where (M^{k})^{∗} denotes the dual of M^{k}, the subspace of M of elements of degree k.

THEOREM 1.6 [LZ1]. The unstable modules H^{∗}V ⊗J(k) are injective in U for all elementary abelian p-groups V and all natural numbers k.

Note that for k = 0 this is just the injectivity of H^{∗}V which was essentially proved in [C], [Mil].

THEOREM 1.7 [LS,S1,S2]. An unstable module N is n-nilpotent iff Hom_{U}(N, H^{∗}V ⊗J(k)) = 0 for all V and all k < n.

\underline{1.8.} In this section we start investigating the quotient category U/Nil_{n} which is defined as follows: The objects of U/Nil_{n} are the same as those of U. For L and M in U/Nil_{n} we have Hom_{U/Nil_{n}}(L, M) = colim Hom_{U}(L′, M/M′), where the colimit is taken over all pairs (L′, M′) where L′ is a submodule of L such that L/L′ is n - nilpotent and M′ is an n - nilpotent submodule of M.

\underline{1.9.} Let I^{<n} be the full subcategory of U^{op} whose objects are the injectives H^{∗}V ⊗ J(k) for all V ∈ E and all k < n and let E_{∞} denote the category of all _{p}−vector spaces. Work of Gabriel [G] suggests to compare U/Nil_{n} with the category _{i}F^{<n} of additive functors from I^{<n} to E_{∞}. We apologize for the heavy notation, but we would like to reserve the symbol F^{<n} for another category which we will introduce below and which is isomorphic to _{i}F^{<n}. The category F^{<n} will be the one that we will work with later. In the case n = 1 which was studied in detail in [HLS1,HLS2], the category I^{<n} can be identified with the _{p}− linearization of the category E and therefore _{i}F^{<n} with the category F of all functors from E to E_{∞}.

\underline{1.10.} We define a functor _{i}f̃^{<n} from U to _{i}F^{<n} by associating to an unstable module M the additive functor whose value on the injective object I is Hom_{U}(M, I)′, the continuous dual of the profinite vector space Hom_{U}(M, I). Here the profinite structure is defined by regarding M as the filtered colimit of its finitely generated A-submodules and hence by regarding Hom_{U}(M, I) as filtered limit of finite dimensional vector spaces. By 1.7 _{i}f̃^{<n} vanishes on the subcategory Nil_{n}, by 1.6 _{i}f̃^{<n} is exact and hence induces a functor from U/Nil_{n} to F^{<n} which we will denote by _{i}f^{<n}. Then it follows from 1.7 again that _{i}f^{<n} is faithful.

\underline{1.11.} In the other direction we define a functor _{i}m^{<n} from _{i}F^{<n} to U as follows. Let l be a fixed integer. Consider the object in _{i}F^{<n} whose value on the injective I is (I^{l})^{∗}. As this is essentially the restriction of the “degree l part” of the dual of the identity functor on U, we will denote it by (Id^{∗})_{l} . Let F be in _{i}F^{<n}. Then the underlying vector space of _{i}m^{<n}(F) in degree l is defined as Hom_{_{i}F^{<n}}((Id^{∗})_{l}), F). The A-module structure of im^{<n}(F) is induced from the A-module structure on the injectives I. Finally _{i}m^{<n} will be the composition of _{i}m^{<n} with the canonical functor from U to U/Nil_{n}. Note that by definition (Id^{∗})_{l} agrees with _{i}f^{<n}(F (l)), F (l) denoting the free unstable module on a generator in degree l. From this one can easily check that _{i}m^{<n} is right adjoint to _{i}f̃^{<n} and that _{i}m^{<n} is right adjoint to _{i}f^{<n}.

\underline{1.12.} We will reformulate 1.9 - 1.11 with the help of the functors T_{V} introduced by the second author [L1,L2]. Recall that T_{V} is a functor from U to U left adjoint to tensoring with H^{∗}V, i.e. there are natural equivalences Hom_{U}(T_{V} M, N) ≅ Hom_{U}(M, H^{∗}V ⊗ N). In particular we see that the value of the functor _{i}f^{<n}(M) on the injective H^{∗}V ⊗ J(k) can be identified with T_{V}^{k} M, the subspace of T_{V} M of elements of degree k. For the convenience of the reader we will recall the following properties of the functor T_{V}. The first one is equivalent to Theorem 1.6.

THEOREM 1.12.1 [L1,L2]. The functor T_{V} is exact.

THEOREM 1.12.2 [L1,L2]. The functor T_{V} commutes with tensor products. More precisely the product on H^{∗}V induces natural isomorphisms T_{V} (M ⊗ N) ⟶ T_{V} M ⊗ T_{V} N for all unstable modules M and N.

PROPOSITION 1.12.3 [L1,L2]. Let V and W be in E. Then there is an isomorphism

T_{V} H^{∗}W ⟶^{≅} ∏_{ρ∈Hom(V,W)} H^{∗}W

whose components are adjoint to the maps H^{∗}W → H^{∗}(V ⊕ W) ≅ H^{∗}V ⊗ H^{∗}W, induced by V ⊕ W → W, (v, w) ↦ (ρv + w).

PROPOSITION 1.12.4 [L1,L2]. Let M be a bounded unstable module (i.e. M is trivial in large degrees) and let V be in E. Then the map T_{V} M → M, adjoint to M ≅ _{p} ⊗ M → H^{∗}V ⊗ M, is an isomorphism.

\underline{1.13.} Let κ_{M,V} : T_{V} M ⟶ H^{∗}V ⊗ T_{V} M be defined as the adjoint of the composition

M ⟶^{γ} H^{∗}V ⊗ T_{V} M ⟶^{ψ⊗id} H^{∗}V ⊗ H^{∗}V ⊗ T_{V} M,

where γ = γ_{M,V} is adjoint to id_{T_{V} M} and ψ = ψ_{V} is the diagonal of the coalgebra H^{∗}V. We observe that κ = κ_{M,V} endows T_{V} M with a natural H^{∗}V - comodule structure. In addition for each α : V → W the induced map T_{α}M : T_{V} M → T_{W} M is a map of H^{∗}V - comodules, if the H^{∗}W - comodule T_{W} M is considered as an H^{∗}V - comodule via the coalgebra map α^{∗} : H^{∗}W → H^{∗}V induced by α. In other words, the following diagram is commutative

T_{V} M  ⟶^{κ_{M,V}}  H^{∗}V ⊗ T_{V} M
↓^{T_{α}M}  ↓^{id⊗T_{α}M}
T_{W} M ⟶^{κ_{M,W}}  H^{∗}W ⊗ T_{W} M  ⟶^{α^{∗}⊗id} H^{∗}V ⊗ T_{W} M .

It is illuminating to think of the functor T_{V} (−) as an algebraic analogue of the mapping space map(BV, −) (cf. [L1,L2]) and of the comodule structure as an analogue of the action of the H - space BV on map(BV, X), given by (x, f) ↦ (y ↦ f(yx)) for x, y ∈ BV, f ∈ map(BV, X).

\underline{1.14.} We formalize this property of T_{V} M and introduce the category F^{<n}. First we note that the category GrE_{∞}^{<n} of graded vector spaces concentrated in degrees between 0 and n − 1 is a monoidal category with product A ⊗_{n} B := (A ⊗ B)^{<n} for A and B in GrE_{∞}. Here we use for a graded vector space C the symbol C^{<n} to denote the graded vector space which in degrees between 0 and n − 1 agrees with C and is trivial in other degrees. Because of this monoidal structure it makes sense to talk about coalgebras and comodules in this category. We will call them n - truncated comodules and n - truncated coalgebras and we will often drop n from the notation if there is no danger of confusion. It is clear that (H^{∗}V)^{<n} is an n - truncated coalgebra and (T_{V} M)^{<n} is an n -truncated comodule over it.

With this terminology the objects of F^{<n} are functors F (not necessarily additive) from the category E to the category U^{<n} of unstable A-modules concentrated in degrees between 0 and n − 1, together with A - linear structure maps Δ_{F} = Δ_{F V} : F V → (H^{∗}V ⊗ F V)^{<n} which define on F V an n - truncated (H^{∗}V)^{<n} - comodule structure; furthermore for each α : V → W the map F α is a map of (H^{∗}V)^{<n} - comodules with F W being considered as an (H^{∗}V)^{<n} - comodule via α^{∗} (cf. 1.13 above). Note that in order to get A - actions on (H^{∗}V)^{<n} (resp. (H^{∗}V ⊗ F V)^{<n}, ...) we have to consider them as quotients rather than as subobjects of H^{∗}V (resp. H^{∗}V ⊗ F V, ...). The morphisms in F^{<n} are all natural transformations which respect these n - truncated comodule structures.

PROPOSITION 1.15. There is a canonical isomorphism of categories _{i}F^{<n} ≅ F^{<n}.

In the sequel we will only work with F^{<n} and the reader could ignore what we have said so far about _{i}F^{<n}. However, it is the category _{i}F^{<n} which links our work with the general theory of localizations in abelian categories [G] and we did not want to suppress this. So we give at least a sketch of the proof.

Note also that for n = 1 both the A - module and comodule structures on F ∈ F^{<n} are necessarily trivial, and hence the proposition gives that F^{<1} identifies with the category F of all functors from E to E_{∞} (cf. 1.9 above).

\underline{Sketch of Proof of Proposition 1.15:} The main point of the proof is that by 1.12.2-1.12.4 we know Hom_{U}(H^{∗}W ⊗ J(m), H^{∗}V ⊗ J(l)), i.e. Hom_{I^{<n}}(H^{∗}V ⊗ J(l), H^{∗}W ⊗ J(m)). Suppose F is in _{i}F^{<n}. For each V ∈ E we define a graded vector space F̃V concentrated in degrees < n by F̃^{k}V = F (H^{∗}V ⊗ J(k)) for k < n. The unstable A - module structure on this graded vector space is induced by the right action of the Steenrod algebra on the collection of the Brown Gitler modules J(k) (cf. [LZ1], [Mil]). To get the comodule structure maps we proceed as follows. As F is in _{i}F^{<n} we get structure maps

Hom_{U}(H^{∗}W ⊗ J(m), H^{∗}V ⊗ J(l)) ⊗ F̃^{l}V ⟶ F̃^{m}W

for all V, W ∈ E and l, m < n which by dualization give maps

F̃^{l}V ⟶ T_{V}^{l} (H^{∗}W ⊗ J(m)) ⊗ F̃^{m}W .

Theorem 1.12.2 and Propositions 1.12.3 , 1.12.4 allow us to identify T_{V} (H^{∗}W ⊗ J(m)) with ∏_{Hom(V,W)} H^{∗}W ⊗ J(m) and then the identification of J(m)^{k} with (F (k)^{m})^{∗} (see [LZ1] for example) gives us maps

F̃^{l}V ⟶ ∏_{Hom(V,W)} ⨁_{k} H^{l−k}W ⊗ Hom(F (k)^{m}, F̃^{m}W)

or all V, W ∈ E and l, m < n. Now it is not difficult to check that the functoriality of F is equivalent to the following three statements

1) These maps (for l fixed and m variable) factor through

∏_{Hom(V,W)} ⨁_{k} H^{l−k}W ⊗ Hom_{U}(F (k), F̃W) ≅ ∏_{Hom(V,W)} ⨁_{k} H^{l−k}W ⊗ F̃^{k}W

and the resulting maps

F̃V ⟶ ∏_{Hom(V,W)}(H^{∗}W ⊗ F̃W)^{<n}

are A - linear.

2) The component corresponding to the identity on V defines an n - truncated (H^{∗}V)^{<n} - comodule structure on F̃ and for each α : V → W the map F α is a homomorphism of truncated (H^{∗}V)^{<n} - comodules as required.

3) F̃V is functorial in V (as unstable A - module).

All these steps can be reversed and hence the preceeding discussion shows also how to reconstruct F from F̃ and we get the isomorphism between the two categories.

\underline{1.16.} With the identification of 1.15 the functor _{i}f^{<n} corresponds to the functor f^{<n} : M ↦ (V ↦ (T_{V} M)^{<n}) and the functor _{i}m^{<n} corresponds to a functor m^{<n} which is right adjoint to f^{<n} by 1.11. above. We will give an alternative description of this adjoint which will be very useful in particular when we will discuss localizations of equivariant cohomology. \underline{1.16.1.} First we recall the “twisted arrow category” C_{♯} of a category C as introduced in [M,p.223] (see also [Q4]). Its objects are the arrows of C and a morphism from the arrow α_{1} to α_{2} is a pair (β_{1}, β_{2}) of arrows such that β_{2}α_{1}β_{1} is defined and equal to α_{2}. There are canonical functors d : C_{♯} → C^{op} (resp. r : C_{♯} → C) given on an object in C_{♯} by taking its domain (resp. range). Furthermore the end of a bifunctor G : C^{op} ×C ⟶ D (which was an important tool in part I of [HLS2]) can be identified with the limit (over C_{♯}) of the functor G ∘ (d, r). In fact, if C is small and D has products and equalizers, then for each functor F : C_{♯} → D there is a canonical isomorphism

lim_{C_{♯}} F ≅ [Eq : ∏_{c∈obC} F (1_{c}) \overrightarrow{⟶} ∏_{α∈morC} F (α)] .

Here the right hand side is the equalizer of the two morphisms between the two products which are induced from the obvious morphisms (in C_{♯}) from 1_{dα} to α and 1_{rα} to α for each α ∈ obC_{♯} = morC (cf. [M,p.223]). If F = G ∘ (d, r) as above, then the equalizer is the end of G; if furthermore G is constant in one of the two variables then the right hand side is the limit of G considered as functor in the remaining variable. In this paper limits over C_{♯} (for certain categories C) will be very important. The categories C that we will be using will not always be small, but they will have obvious small skeleta and we will work with such skeleta whenever necessary. \underline{1.16.2.} Now let F be in F^{<n}. Associated to F we have two bifunctors F_{1} and F_{2} from E^{op} × E to U, given by F_{1}(V, W) = H^{∗}V ⊗ F W and F_{2}(V, W) = H^{∗}V ⊗ (H^{∗}V ⊗ F W)^{<n}. If we consider these functors as functors on E_{♯} via composition with (d, r) as above and for simplicity denote the resulting functors again by F_{1} and F_{2}, then we have the following two natural transformations μ and ν from F_{1} to F_{2}. In fact μ is actually a transformation of bifunctors and μ(V, W) is the truncation of the map ψ_{V} ⊗ id_{F W} (with ψ_{V} as in 1.13 above). However, for ν we have to give up the bifunctoriality to get a natural transformation. If α is a morphism from V to W in E, i.e. α ∈ obE_{♯} then ν(α) is the composition of id_{H^{∗}V} ⊗ Δ_{F W} followed by id_{H^{∗}V} ⊗ (H^{∗}α ⊗ id_{F W})^{<n}. Now the equalizer of μ and ν is a functor from E_{♯} to U^{<n} denoted by EqF and we claim that the functor F^{<n} → U, F ↦ lim_{E_{♯}} EqF is right adjoint to f^{<n}. To see this we observe first that for F, G ∈ F^{<n} we can identify Hom_{F^{<n}}(F, G) with lim_{α∈E_{♯}} Hom_{U−H^{∗}dα}(F dα, Grα) where Hom_{U−H^{∗}dα}(F dα, Grα) denotes the A-linear maps from F dα to Grα which are also H^{∗}dα - comodule maps if Grα is considered as H^{∗}dα - comodule as in 1.14. Then we get the following sequence of natural isomorphisms (use that T_{dα} is left adjoint to tensoring with H^{∗}dα to get the second isomorphism)

Hom_{F^{<n}}(f^{<n}M, F) ≅ lim_{α∈E_{♯}} Hom_{U−H^{∗}dα}((f^{<n}M)dα, F rα) ≅

≅ lim_{α∈E_{♯}} Hom_{U}(M, (EqF)α) ≅ Hom_{U}(M, lim_{α∈E_{♯}}(EqF)α) .

\underline{1.16.3.} As an example consider the following special cases. Let F be in F^{<1}. Then (EqF)α = H^{∗}dα ⊗ F rα and m^{<1}F is the end of the bifunctor H^{∗} ⊗ F, i.e it can be identified with the unstable A -module m(F) which was studied in [HLS1,HLS2]. More generally consider Σ^{n−1}F ∈ F^{<n} (with necessarily trivial A - module and truncated comodule structure). Then we get (EqΣ^{n−1}F)α = H^{∗}dα ⊗ Σ^{n−1}F rα and hence m^{<n}Σ^{n−1}F ≅ Σ^{n−1}m^{<1}F.

Below we will always work with F^{<n}, f^{<n} and m^{<n}. In case n = 1 we will also use the notation F, f and m in accordance with [HLS1,HLS2].

\underline{1.17.} We define a functor F in F^{<n} to be analytic if all its homogeneous components are analytic in the sense of [HLS1,HLS2]. The full subcategory of analytic functors in F^{<n} will be denoted by F_{ω}^{<n}. The Taylor functor tF of an arbitrary F in F^{<n} is defined as the largest analytic subfunctor of F. In fact, we claim that the “degree n part” of tF is just the Taylor functor of F^{n} in the sense of [HLS1,HLS2], i.e. (tF)^{n} = t(F^{n}). To see that the “comodule - structure” on F is inherited by tF we observe that by dualization it suffices that the maps H_{k}V ⊗ F^{l}V ⟶ F^{l−k}V (k ≤ l < n) induce such maps on tF. Now H_{k} is clearly analytic (in fact polynomial of degree k) and hence by passing to Taylor functors we get such natural transformations H_{k} ⊗ tF^{l} ↪ t(H_{k} ⊗ F^{l−k}) ⟶ tF^{l−k} for all k ≤ l < n. We leave it to the reader to verify that tF (V) is an unstable A - module for each V ∈ E.

PROPOSITION 1.18. The functor f^{<n} takes values in F_{ω}^{<n}. \underline{Proof:} As the functor f^{<n} commutes with colimits and F_{ω}^{<n} is closed under forming colimits it suffices to check the case of the free unstable modules F (n). However

T_{V} F (n) ≅ ⨁_{k=1}^{n} F (k) ⊗ H_{n−k}V

is clearly analytic and hence we are done.

2. The equivalence between U/Nil_{n} and F_{ω}^{<n}

In this section we will prove the following result.

THEOREM 2.1. The functors f^{<n} and m^{<n} define mutually inverse equivalences between the categories U/Nil_{n} and F_{ω}^{<n}.

Observe that Theorem 2.1 identifies the quotient category Nil_{n−1}/Nil_{n} with the full subcategory of F_{ω}^{<n} consisting of functors concentrated in degree n − 1 which in turn identifies up to (n − 1) - fold suspension with F_{ω}. In particular this shows that the quotient categories Nil_{n−1}/Nil_{n} are all equivalent [S1,S2].

Our strategy for proving 2.1 is the same as in the case n = 1 [HLS2]. As there we denote by η : 1_{U} → m^{<n}f^{<n} (resp. ϵ : f^{<n}m^{<n} → 1_{F^{<n}}) the unit (resp. counit) of the adjunction between f^{<n} and m^{<n}. Then the key result reads as follows (cf. [HLS2, Thm.I.7.3]).

THEOREM 2.2. Let F be in F^{<n}. Then the natural morphism ϵ_{F} : f^{<n}m^{<n}F ⟶ F induces an isomorphism between f^{<n}m^{<n}F and the Taylor functor tF of F.

Theorem 2.2 proves half of Theorem 2.1. The other half is given by the following formal consequence of Theorem 2.2 (cf. [HLS2, I.7.]).

PROPOSITION 2.3. Let M be an unstable A- module. Then the kernel and cokernel of the natural morphism η_{M} : M ⟶ m^{<n}f^{<n}M are n - nilpotent.

The remainder of this section is devoted to the proof of Theorem 2.2.

\underline{2.4.} Let F be in F^{<n}. First we show that we have a “commutative diagram” with exact rows

f^{<n}m^{<n}F ⟶ f^{<n} lim_{α∈E_{♯}} F_{1}α \overrightarrow{⟶}_{ν}^{μ} f^{<n} lim_{α∈E_{♯}} F_{2}α
↓_{ϵ_{F}}  ↓  ↓
F  ⟶ lim_{α∈E_{♯}} f^{<n}F_{1}α \overrightarrow{⟶}_{ν}^{μ} lim_{α∈E_{♯}} f^{<n}F_{2}α

in which F_{1}, F_{2} are as in 1.16.2, the second and third vertical arrows are the obvious ones (induced by f^{<n}) and in which ϵ̃_{F} is induced by these arrows. The natural transformations μ and ν from 1.16 induce the morphisms in this diagram which by abuse of notation are again labelled μ and ν. Similar abuse of notation will be tacitly used below. Finally exactness means that the left hand term in each row identifies with the equalizer of the corresponding two right hand horizontal arrows and commutativity refers to the two diagrams with μ and ν separately. In fact, for the top line exactness is an immediate consequence of exactness of f^{<n} and the straightforward isomorphism lim_{α}(EqF)α ≅ [Eq : lim_{α} F_{1}α \overrightarrow{⟶}_{ν}^{μ} lim_{α} F_{2}α].

Here and in the following Eq : A \overrightarrow{⟶} B will always denote the equalizer of the two arrows

under consideration.

Now consider exactness of the bottom line. If S is a set we denote by _{p}^{S} the algebra of all functions from S to _{p}. From 1.12.2 - 1.12.4 we get for each α ∈ E_{♯}

(f^{<n}(F_{1}α))(−) ≅ _{p}^{Hom(−,dα)} ⊗ (H^{∗}dα ⊗ F rα)^{<n}

and

(f^{<n}(F_{2}α))(−) ≅ _{p}^{Hom(−,dα)} ⊗ (H^{∗}dα ⊗ H^{∗}dα ⊗ F rα)^{<n} .

Now observe that for a ring R with unit and an R module M the strucure map R⊗M ⟶ M induces an isomorphism between M and the coequalizer of the two canonical maps from R ⊗ R ⊗ M to R ⊗ M. Similarly in our case the map F rα ⟶ (H^{∗}dα ⊗ F rα)^{<n} which defines the truncated (H^{∗}dα)^{<n} - comodule structure on F rα induces an isomorphism between F rα and

Eq : (H^{∗}dα ⊗ F rα)^{<n} \overrightarrow{⟶}_{ν(α)}^{μ(α)} (H^{∗}dα ⊗ H^{∗}dα ⊗ F rα)^{<n}

and hence, using again that lim_{α} and Eq commute, we identify the “bottom line equalizer” with

lim_{α∈E_{♯}}[Eq : (f^{<n}(F_{1}α))(−) \overrightarrow{⟶}_{ν(α)}^{μ(α)} (f^{<n}(F_{2}α))(−)] ≅ lim_{α∈E_{♯}} _{p}^{Hom(−,dα)} ⊗ F rα ≅ F (−) .

The commutativity of the two diagramms is obvious. We leave it to the reader to verify that after these identifications the morphisms ϵ̃_{F} and ϵ_{F} agree (hint: use the description of the adjunction between f^{<n} and m^{<n} given in 1.16.).

The proof of Theorem 2.2 is now an easy consequence of the following two results.

LEMMA 2.5. The functor f^{<n} induces an isomorphism

(f^{<n} lim_{α∈E_{♯}} F_{1}α)(−) ⟶ lim_{α∈E_{♯}} _{p}^{Hom(−,dα)} ⊗ (H^{∗}dα ⊗ tF (rα))^{<n} .

LEMMA 2.6. The functor f^{<n} induces a monomorphism

(f^{<n} lim_{α∈E_{♯}} F_{2}α)(−) ⟶ lim_{α∈E_{♯}} _{p}^{Hom(−,dα)} ⊗ (H^{∗}dα ⊗ H^{∗}dα ⊗ F rα)^{<n} .

In fact with a little more effort one can show that in Lemma 2.6 the image is also given by replacing F by tF. However, for the proof of Theorem 2.2 the given version of Lemma 2.6 will be good enough. Before we turn towards the proofs of 2.5 and 2.6 we give now the proof of 2.2.

\underline{Proof of Theorem 2.2:} From 2.4, 2.5 and 2.6 we get an isomorphism (induced by ϵ̃_{F}) between f^{<n}m^{<n}F and

Eq : lim_{α∈E_{♯}} _{p}^{Hom(−,dα)} ⊗ (H^{∗}dα ⊗ tF (rα))^{<n} \overrightarrow{⟶}_{ν}^{μ} lim_{α∈E_{♯}} _{p}^{Hom(−,dα)} ⊗ (H^{∗}dα ⊗ H^{∗}dα ⊗ F rα)^{<n}

≅ lim_{α∈E_{♯}} Eq[ _{p}^{Hom(−,dα)} ⊗ (H^{∗}dα ⊗ tF (rα))^{<n} \overrightarrow{⟶}_{ν(α)}^{μ(α)} _{p}^{Hom(−,dα)} ⊗ (H^{∗}dα ⊗ H^{∗}dα ⊗ F rα)^{<n}].

Because μ(α) and ν(α) factor through the inclusion

_{p}^{Hom(−,dα)} ⊗ (H^{∗}dα ⊗ H^{∗}dα ⊗ tF (rα))^{<n} ⟶ _{p}^{Hom(−,dα)} ⊗ (H^{∗}dα ⊗ H^{∗}dα ⊗ F rα)^{<n} ,

we can identify their equalizer with _{p}^{Hom(−,dα)} ⊗ tF (rα) (by using the truncated comodule structures as in 2.4 above) and hence we get

f^{<n}m^{<n}F (−) ≅ lim_{α∈E_{♯}} _{p}^{Hom(−,dα)} ⊗ tF (rα) ≅ tF (−) .

\underline{2.7.} For the proof of 2.5 and 2.6 we recall the primitive filtration on H^{∗}V and generalize it to any functor G : E^{op} → E_{∞} (cf. [HLS2, I.6, I.7]).

We set P_{m}G(V) = Ker(GV ⟶^{G(∇_{m+1}^{V})} G(V^{⊕(m+1)})), where ∇_{m+1}^{V} is the following morphism in _{p}[E], the _{p} - linearization of the category E: If q_{k} : V^{⊕(m+1)} → V denotes the kth projection and [q_{k}] the corresponding element in the group algebra _{p} Hom(V^{⊕(m+1)}, V) then

∇_{m+1}^{V} := ([q_{1}] − [0])([q_{2}] − [0]) …([q_{m+1}] − [0]) ∈ Hom_{_{p}[E]} (V^{⊕(m+1)}, V).

With these definitions it is clear that

(2.7.1)  f^{<n}(P_{m}H^{∗}V)(−) = P_{m}(f^{<n}H^{∗}V)(−)

(In this formula P_{m} is applied degreewise). Similarly we define for any functor G : E → E_{∞} a new functor Q_{m}G via Q_{m}G(V) =

Coker(G(V^{⊕(m+1)}) ⟶^{G(∇_{m+1}^{V})} GV).

\underline{2.8. Proof of Lemma 2.5:} We approximate H^{∗}V via the primitive filtration, i.e. we write H^{∗}V = colim_{m} P_{m}H^{∗}V. Then we clearly get

f^{<n}( lim_{α∈E_{♯}} F_{1}α) = f^{<n}( lim_{α∈E_{♯}} colim_{m} P_{m}H^{∗}dα ⊗ F rα) = f^{<n}(colim_{m} lim_{α∈E_{♯}} P_{m}H^{∗}dα ⊗ F rα) .

The functor f^{<n} commutes with arbitrary colimits and we obtain

f^{<n}( lim_{α∈E_{♯}} F_{1}α) ≅ colim_{m} f^{<n}( lim_{α∈E_{♯}} P_{m}H^{∗}dα ⊗ F rα) .

Although f^{<n} does not commute with limits in general we see as in [HLS2, I.7.6] that in the case of lim_{α} P_{m}H^{∗}dα ⊗ F rα it does commute. Now using 1.12.2 - 1.12.4 and (2.7.1) above we get for each U ∈ E

f^{<n}( lim_{α∈E_{♯}} F_{1}α)(U) ≅ colim_{m} lim_{α∈E_{♯}}(P_{m}( _{p}^{Hom(U,dα)} ⊗ H^{∗}dα) ⊗ F rα)^{<n} .

Next we consider a fixed degree k, 0 ≤ k < n. By dualizing we can rewrite

lim_{α}(P_{m}( _{p}^{Hom(U,dα)} ⊗ H^{∗}dα) ⊗ F rα)^{k} ≅ ⨁_{j} Hom_{F}(Q_{m}( _{p} Hom(U, −) ⊗ H_{j}(−)), F^{k−j}(−)) .

Then Lemma 2.10, 2.11 and 2.12 below show that the right hand side can be identified with ⨁_{j} Hom_{F}( _{p} Hom(U, −) ⊗ H_{j}(−), tF^{k−j}(−)) and dualizing back completes the proof.

\underline{2.9. Proof of Lemma 2.6:} As in the proof of Lemma 2.5 we get for each U ∈ E

f^{<n}( lim_{α∈E_{♯}} F_{2}α)(U) ≅ colim_{m} lim_{α∈E_{♯}}(P_{m}( _{p}^{Hom(U,dα)} ⊗ H^{∗}dα ⊗ H^{∗}dα) ⊗ F rα)^{<n}

and this is clearly contained in lim_{α∈E_{♯}}( _{p}^{Hom(U,dα)} ⊗ H^{∗}dα ⊗ H^{∗}dα ⊗ F rα)^{<n}.

Before we get to the proof of 2.10 - 2.12 we recall from [HLS2] that t_{m} : F → F denotes the functor which associates to F ∈ F the largest polynomial subfunctor of degree m. Furthermore the functor V ↦ _{p} Hom(U, V) which represents U ↦ F U will be denoted by P_{U}. A functor will be called finitely generated if it is a homomorphic image of a finite direct sum of functors P_{U_{i}}.

LEMMA 2.10. Q_{m} is left adjoint to t_{m}.

LEMMA 2.11. The functors P_{U} ⊗ H_{k} are all finitely generated.

LEMMA 2.12. If F is finitely generated and G is arbitrary then the natural inclusion

colim_{m} Hom_{F}(F, t_{m}G) ↪ Hom_{F}(F, tG)

is an isomorphism.

\underline{Proof of Lemma 2.10:} This is implicit in the proof given in [HLS2] in the case n = 1 and follows immediately from the different characterizations of a functor of degree m given in [HLS2, I.6.7].

\underline{Proof of Lemma 2.11:} It is not difficult to check that H_{k} is finitely generated. The lemma follows because P_{U} ⊗ P_{V} is isomorphic to P_{U⊕V}.

\underline{Proof of Lemma 2.12:} This follows immediately from the fact that P_{U} represents the functor F ↦ F U.

3. Localizing away from Nil_{n}; the invariants d_{0} and d_{1}

DEFINITION 3.1. a) An unstable module M is called Nil_{n} - reduced iff Hom_{U}(N, M) = 0 for all n -nilpotent modules N.

b) An unstable module M is called Nil_{n} - closed iff Ext_{U}^{i}(N, M) = 0 , i = 0, 1, for all n - nilpotent modules N.

It is easy to see that M is Nil_{n} - closed if and only if Hom_{U}(φ, M) is an isomorphism for each morphism φ with kernel and cokernel in Nil_{n}. Also note that Nil_{n} - reduced (resp. Nil_{n} - closed ) in the sense of this definition is Nil_{n−1} - reduced (resp. Nil_{n−1} - closed) in the sense of [BZ]. We refer to [BZ] for a discussion of numerous equivalent characterizations of Nil_{n} - reduced resp. Nil_{n} - closed modules. Instead of Nil_{1} - reduced (resp. Nil_{1} - closed) we will also say Nil - reduced (resp. Nil - closed) in accordance with [HLS1,HLS2].

\underline{3.2.Examples} a) By 1.6 and (the easy direction of) 1.7 H^{∗}V ⊗ J(k) is Nil_{n} - closed for each V ∈ E if n > k. Clearly H^{∗}V ⊗ J(k) is not Nil_{k} - reduced.

b) If 0 → M_{1} → M_{2} → M_{3} is an exact sequence with M_{2} being Nil_{n} - closed and M_{3} being Nil_{n} - reduced, then M_{1} is Nil_{n} - closed.

c) By (a) and (b) H^{∗}V ⊗ B is Nil_{n} - closed for each B ∈ U which is trivial in degrees ≥ n.

d) If M is Nil - closed, then Σ^{n−1}M is Nil_{n} - closed.

e) If F is in F^{<n} then m^{<n}F is Nil_{n} - closed (because m^{<n} is right adjoint to f^{<n} and f^{<n} vanishes on Nil_{n}).

\underline{3.3.} The subcategories Nil_{n} of U are localizing in the sense of [G], so there is a functor L_{n} : U → U and a natural transformation λ_{n} : 1_{U} → L_{n} such that

(1) L_{n}M is Nil_{n} - closed for each M ∈ U

(2) λ_{n,M} has n-nilpotent kernel and cokernel.

Note that (1) and (2) imply that Ker λ_{n,M} is the largest n - nilpotent submodule of M. Such a natural transformation is unique up to isomorphism. We call L_{n}M or more precisely λ_{n}M the localization of M away from Nil_{n} and L_{n} the localization functor. The localization functor is left exact but not exact in general (cf. II.5.3.2 and II.5.4.5 below for examples). Clearly M is Nil_{n}-reduced resp. Nil_{n}-closed iff λ_{n}M is a monomorphism resp. isomorphism.

By Proposition 2.3 and Example 3.2.(e) above the map η_{M} : M ⟶ m^{<n}f^{<n}M is localization away from Nil_{n} and the exactness of f^{<n} and the left exactness of m^{<n} (m^{<n} being a right adjoint) shows again that L_{n} is left exact. By using the description of m^{<n}F given in 1.16 we arrive at the following

SCHOLIUM 3.4. The morphism

M ⟶ lim_{α∈E_{♯}}[Eq : H^{∗}dα ⊗ f^{<n}M(rα) \overrightarrow{⟶}_{ν(α)}^{μ(α)} H^{∗}dα ⊗ (H^{∗}dα ⊗ f^{<n}M(rα))^{<n}] ,

induced by the maps λ(α) := M ⟶^{γ_{M,rα}} H^{∗}rα ⊗ f^{<n}M(rα) ⟶^{α^{∗}⊗id} H^{∗}dα ⊗ f^{<n}M(rα), is

localization away from Nil_{n} for any unstable module M.

By abuse of notation we have denoted the “truncation” of the map γ_{M,rα} introduced in 1.13 again by γ_{M,rα}.

DEFINITION 3.5. Let M be an unstable module. We define

d_{0}M := inf{k ∈ |M is Nil_{k+1} − reduced}
d_{1}M := inf{k ∈ |M is Nil_{k+1} − closed}

(By convention the infimum of the empty set is ∞.) In terms of localizations we see that

d_{0}M := inf{k ∈ |λ_{k+1}M is mono}
d_{1}M := inf{k ∈ |λ_{k+1}M is iso}.

Here we interpret λ_{∞} as inverse limit of the λ_{k}’s: this is always an isomorphism because λ_{k} is an isomorphism in degrees < k.

We list some properties of these invariants.

PROPOSITION 3.6.

a) d_{0}(H^{∗}V ⊗ J(k)) = d_{1}(H^{∗}V ⊗ J(k)) = k. b) Let 0 ⟶ M_{1} ⟶ M_{2} ⟶ M_{3} be an exact sequence of unstable modules. Then d_{0}M_{1} ≤ d_{0}M_{2}, d_{1}M_{1} ≤ max{d_{1}M_{2}, d_{0}M_{3}}.

c) Let 0 ⟶ M_{1} ⟶ M_{2} ⟶ M_{3} ⟶ 0 be an exact sequence of unstable modules. Then d_{0}M_{2} ≤ max{d_{0}M_{1}, d_{0}M_{3}}, d_{1}M_{2} ≤ max{d_{1}M_{1}, d_{1}M_{3}} and d_{0}M_{3} ≤ max{d_{0}M_{2}, d_{1}M_{1}} d) d_{i}(M_{1} ⊕ M_{2}) = max{d_{i}M_{1}, d_{i}M_{2}} for i = 0, 1 and any unstable modules M_{1}, M_{2}.

e) d_{0}(M ⊗ N) = d_{0}M + d_{0}N and d_{1}(M ⊗ N) = max{d_{0}M + d_{1}N, d_{1}M + d_{0}N} for any nontrivial unstable modules M_{1}, M_{2}.

\underline{Proof:} (a) - (d) are straightforward; (e) is just a restatement of Proposition 1.17 in [BZ].

We will see in I.4 below that in many interesting cases d_{0}M and d_{1}M are both finite and give interesting invariants of M. The study of these invariants in the case of equivariant cohomology will be the subject of sections II.1-II.3.

\underline{3.7.} Next we consider the sequence of localizations λ_{n}, n = 1, 2, .... A Nil_{n−1}-closed module M is clearly Nil_{n}-closed, so if we identify L_{n−1}M with L_{n}L_{n−1}M via λ_{n}(L_{n−1}M) and set

τ_{n−1} = L_{n}λ_{n−1} : L_{n} ⟶ L_{n}(L_{n−1}) ≅ L_{n−1}

we get a factorization λ_{n−1} = τ_{n−1}λ_{n}. We will be interested in the following tower of localizations:

.  .  .
.  .  .
.  .  .
M ⟶^{λ_{3}} L_{3}M
∥  ↓^{τ_{2}}
M ⟶^{λ_{2}} L_{2}M
∥  ↓^{τ_{1}}
M ⟶^{λ_{1}} L_{1}M

By left exactness of L_{n} and definition of τ_{n} we see that both λ_{k}M and τ_{k}M are mono for k > d_{0}M.

\underline{3.8.} It is clear that Ker λ_{n−1,M}/ Ker λ_{n,M} injects into Ker τ_{n−1,M} and that we can identify f^{<n}(Ker λ_{n−1,M}/ Ker λ_{n,M}) with f^{<n} Ker τ_{n−1,M} via this injection. In addition Ker τ_{n−1,M} = 0 if and only if f^{<n} Ker τ_{n−1,M} = 0 (because Ker τ_{n−1,M} is a submodule of a Nil_{n} - closed module and hence does not contain non-trivial n - nilpotent submodules). In particular we see that Ker λ_{n−1,M} = Ker λ_{n,M} if and only if f^{<n} Ker τ_{n−1,M} = 0. So studying the maps τ_{k}, k = 1, 2, ... is closely related to studying the decreasing sequence of submodules Ker λ_{k}, k = 1, 2, ... .

Furthermore this tower is also useful for computations. The point is that, although one can compute L_{n}M (at least in principle) if f^{<n}M is known, such a computation is often not very illuminating and tends to give an inconvenient description of L_{n}M (the reader should try to compute L_{n}H^{∗}V for n > 1 in this way to become convinced of this). It is often better to try to describe L_{n}M inductively. This observation gives additional impetus to turn attention towards analyzing the maps τ_{n,M}. Although our analysis will not be complete it will be helpful in particular cases (cf. II.5).

We consider the map f^{<n}τ_{n−1} : f^{<n}L_{n}M ⟶ f^{<n}L_{n−1}M. The kernel and cokernel of this map are concentrated in degrees n − 1 and we can write Ker f^{<n}τ_{n−1} = Σ^{n−1}k_{n}M, Coker f^{<n}τ_{n−1} = Σ^{n−1}c_{n}M with functors k_{n}M, c_{n}M in F_{ω}. Applying the left exact functor m^{<n} and identifying m^{<n}Σ^{n−1} with Σ^{n−1}m (cf.1.16) we get the following sequence of unstable modules:

0 ⟶ Σ^{n−1}mk_{n}M ⟶ L_{n}M ⟶^{τ_{n−1}} L_{n−1}M ⟶ Σ^{n−1}mc_{n}M

in which the composition of two successive arrows are all trivial and which is exact except possibly at L_{n−1}M; in particular we see that Ker τ_{n−1} ≅ Σ^{n−1}mk_{n}M. If k_{n}M = 0, then we also have exactness at L_{n−1}M.

\underline{3.9.} The preceeding discussion used that f^{<n}L_{n−1}M is determined by f^{<n}M, but we did not make this explicit. We will now give a more direct description of Ker τ_{n−1} in terms of f^{<n}M. For this we consider for each α ∈ E_{♯} the following “commutative” diagram in which the second and third row are exact. (Here commutativity and exactness are to be understood as in 2.4 and μ, ν are the natural transformations introduced in 1.16)

H^{∗}dα ⊗ (f^{<n}M(rα))^{n−1} \overrightarrow{⟶}_{ν(α)}^{μ(α)} H^{∗}dα ⊗ (H^{∗}dα ⊗ f^{<n}M(rα))^{n−1}
↓  ↓
Eqf^{<n}M(α) ⟶  H^{∗}dα ⊗ f^{<n}M(rα)  \overrightarrow{⟶}_{ν(α)}^{μ(α)}  H^{∗}dα ⊗ (H^{∗}dα ⊗ f^{<n}M(rα))^{<n}
↓  ↓  ↓
Eqf^{<n−1}M(α) ⟶ H^{∗}dα ⊗ f^{<n−1}M(rα) \overrightarrow{⟶}_{ν(α)}^{μ(α)} H^{∗}dα ⊗ (H^{∗}dα ⊗ f^{<n−1}M(rα))^{<n−1}

The diagram shows that the kernel of Eqf^{<n}M(α) ⟶ Eqf^{<n−1}M(α) is isomorphic to H^{∗}dα ⊗ P^{n−1}f^{<n}M(α) where P^{n−1}f^{<n}M(α) denotes the primitives in degree (n − 1) of the truncated (H^{∗}dα)^{<n} - comodule f^{<n}M(rα). This time we can write P^{n−1}f^{<n}M(α) = Σ^{n−1}p_{n−1}M(α) where p_{n−1}M is in E_{∞}^{E_{♯}}. Then we get by left exactness of m^{<n}:

Ker τ_{n−1} ≅ Ker(m^{<n}f^{<n}M ⟶ m^{<n−1}f^{<n−1}M) ≅ Σ^{n−1} lim_{α} H^{∗}dα ⊗ p_{n−1}M(α) .

Next we have a canonical identification

lim_{α∈E_{♯}} H^{∗}dα ⊗ p_{n−1}M(α) ≅ Hom_{E_{∞}^{E_{♯}}} (H_{∗}r, p_{n−1}M) .

where the A - module structure for the last term is induced from the right A - action on H_{∗}. (Note that H_{∗}r and p_{n−1}M are both covariant functors from E_{♯} to E_{∞} while H_{∗}d is contravariant.) The functor F = E_{∞}^{E} ⟶ E_{∞}^{E_{♯}}, induced by precomposition with r : E_{♯} → E, has a right adjoint (by the theory of Kan extensions) denoted by r̃ and finally we get

Ker τ_{n−1} ≅ Σ^{n−1} Hom_{F}(H_{∗}, r̃p_{n−1}M) .

In fact, it is not hard to check that r̃p_{n−1}M can be explicitly described as follows: For V ∈ E we have

r̃p_{n−1}M(V) = ⋂_{V →^{α}W} Ker[(f^{<n}M)^{n−1}(V) ⟶^{α_{∗}} (f^{<n}M)^{n−1}(W) ⟶ (H̃^{∗}W ⊗ f^{<n}M(W))^{n−1}]

where α_{∗} is induced by α and the second arrow is the composition of the comodule structure map in degree n − 1 followed by the canonical projection of H^{∗}W ⊗ (f^{<n}M)^{n−1}(W) onto H̃^{∗}W ⊗ (f^{<n}M)^{n−1}(W). As a subfunctor of f^{<n}M the functor r̃p_{n−1}M is analytic and hence agrees with k_{n}M because both functors produce via Σ^{n−1}m the same unstable module Ker τ_{n−1}.

4. The case of finitely generated unstable modules over unstable noetherian algebras

\underline{4.1.} We recall that K denotes the category of unstable algebras. If K is a fixed unstable algebra then K − U denotes the category whose objects are unstable modules M together with a K - module structure such that the action K ⊗ M → M is A - linear. Morphisms in K − U are homomorphisms which are both A - and K - linear. Finally the full subcategory of K − U consisting of those objects which are finitely generated as K - modules is denoted by K_{fg} − U.

\underline{4.2.} Next recall that the functors T_{V} : U → U induce functors from K to itself still denoted by T_{V} [L1,L2]. From this one deduces without difficulty that the localization map λ_{n,K} :

K → L_{n}K is a homomorphism of unstable algebras for each K ∈ K. (For a more elementary proof of this statement see [BZ].)

Furthermore for fixed K ∈ K the functor T_{V} induces a functor from K − U to T_{V} K − U. From this we deduce easily that L_{n} induces a functor from K − U to L_{n}K − U and, if we pull back the L_{n}K - module structure via λ_{n,K}, then λ_{n} induces a natural transformation 1_{K−U} ⟶ L_{n}.

We remark that it is straightforward to deduce from the equivalence U/Nil_{n} ≅ F_{ω}^{<n} of section 2 an analogous equivalence between an appropriately defined quotient categoriy K/Nil_{n} and a suitable functor category; the same remark applies to the case of K − U. We leave the details to the interested reader.

In the following we will consider the case of M ∈ K_{fg} − U for a noetherian algebra K more closely. Note that the transcendence degree of a noetherian algebra (i.e. the maximal number of algebraically independant homogeneous elements) is always finite. We denote this number by d(K). The following result makes this case particularly interesting.

THEOREM 4.3 [He]. Suppose K is noetherian of finite transcendence degree ≤ d and suppose M is in K_{fg} − U. Then there is an exact sequence

0 ⟶ M ⟶ I_{0} ⟶ I_{1}

such that I_{0} and I_{1} are finite direct sums of objects H^{∗}V_{α} ⊗ J(n_{α}) with dimV_{α} ≤ d for all α. In particular, M is Nil_{n} - closed if n is larger than all of the n_{α} and the invariants d_{0}M, d_{1}M are finite.

As a consequence such an M is determined by f^{<n}M and the localization away from Nil_{n} is an isomorphism if n is as in the theorem. In fact, in this case f^{<n}M is already determined by its value on V_{d} := ( /p)^{d} as the following result shows. Note that for each V ∈ E the monoid EndV_{d} acts on the set Hom(V, V_{d}) by composition and on T_{V_{d}}M by naturality.

PROPOSITION 4.4. Suppose M is an unstable module and that there is an exact sequence 0 ⟶ M ⟶ I_{0} ⟶ I_{1} such that I_{0} and I_{1} are arbitrary direct sums of objects H^{∗}V_{α} ⊗ J(n_{α}) with dimV_{α} ≤ d for all α. Then the natural map

T_{V} M ⟶ Hom_{EndV_{d}}(Hom(V, V_{d}), T_{V_{d}}M)

is an isomorphism.

\underline{Proof:} First we observe that the result is true if M = H^{∗}V_{α}⊗J(n_{α}) by an explicit computation using 1.12.2-1.12.4. The general case follows because both sides are additive and left exact in M.

Now we consider the one-object category EndV_{d} associated to the monoid EndV_{d}. This category is a full subcategory of E and the inclusion of this subcategory yields by Proposition 4.4 the following simplification of our description of the localization away from Nil_{n} given in 3.4.

PROPOSITION 4.5. Asume that M is an unstable module and that there is an exact sequence 0 ⟶ M ⟶ I_{0} ⟶ I_{1} such that I_{0} and I_{1} are arbitrary direct sums of objects H^{∗}V_{α} ⊗ J(n_{α}) with dimV_{α} ≤ d for all α. Then the morphism

M ⟶ lim_{α∈(EndV_{d})_{♯}}[Eq : H^{∗}V_{d} ⊗ f^{<n}M(V_{d}) \overrightarrow{⟶}_{ν(α)}^{μ(α)} H^{∗}V_{d} ⊗ (H^{∗}V_{d} ⊗ f^{<n}M(V_{d}))^{<n}] ,

induced by the maps λ(α) : M ⟶^{γ_{M,V_{d}}} H^{∗}V_{d} ⊗ f^{<n}M(V_{d}) ⟶^{α^{∗}⊗id} H^{∗}V_{d} ⊗ f^{<n}M(V_{d}) for α ∈

mor(EndV_{d}), is localization away from Nil_{n}.

The remark after 3.4 concerning γ_{M,V_{d}} applies here as well.

We remark that this result can be proven without using section 2. The point is that the limit is taken over a finite category, so f^{<n} commutes with the limit and hence it is easy to verify that the map from M to the limit induces an isomorphism after applying f^{<n}. That the limit is Nil_{n} - closed follows from 3.2.(b) and 3.2.(c).

In the remainder of this section we will give still another description of the localization away from from Nil_{n} in case K is noetherian and M is in K_{fg} − U.

\underline{4.6.} To an unstable algebra K we associate a category S(K) as follows. Its objects are the morphisms of unstable algebras φ : K → H^{∗}V for V ∈ E; sometimes it will be convenient to denote these objects by pairs (V, φ). Then the set of morphisms from (V_{1}, φ_{1}) to (V_{2}, φ_{2}) are all homomorphisms V_{1} ⟶^{α} V_{2} such that φ_{1} = α^{∗}φ_{2}. Let R(K) be the full subcategory of S(K) consisting of those objects φ : K → H^{∗}V for which K becomes a finitely generated K - module via φ. For any morphism φ : K → H^{∗}V in S(K) there is a unique maximal subspace of V, denoted kerφ, and a unique morphism φ̄ : K → H^{∗}(V/kerφ) in S(K) such that φ factors as π_{φ}^{∗}φ̄ with π_{φ} denoting the projection V → V/kerφ. For the elementary properties of this construction we refer the reader to [HLS2]. Now let K be noetherian. Then R(K) is a category with a finite skeleton and is the opposite of Rector’s category introduced in [R]. In this case an object φ : K → H^{∗}V of S(K) is in R(K) iff kerφ = 0 and the assignment (V, φ) ↦ (V/kerφ, φ̄) defines a functor reg : S(K) → R(K); the crucial point is that for K noetherian we have the formula ker(α^{∗}φ) = α^{−1}(kerφ). Furthermore we have reg ∘ i = id_{R(K)} (i denoting the inclusion of R(K) into S(K)) and the maps π_{φ} : V → V/kerφ define a natural transformation π : id_{S(K)} → i ∘ reg. Finally precomposition with π gives a natural isomorphism Hom_{R(K)}(regφ, ψ) ≅ Hom_{S(K)}(φ, iψ) for each φ ∈ S(K) and ψ ∈ R(K), i.e. the functors reg and i form an adjoint pair.

\underline{4.7.} Now we turn towards the functors T_{V}. Let K be an unstable algebra. Then T_{V} K is an unstable algebra and we can consider its connected components T_{V} (K; φ) (with φ running through Hom_{K}(K, H^{∗}V)) which are defined as T_{V} (K; φ) := _{p}(φ) ⊗_{T_{V}^{0} K} T_{V} K where _{p}(φ) denotes _{p} considered as a module over T_{V}^{0} K via the adjoint of φ. Similarly, if M is in K − U then T_{V} M is in T_{V} K − U and one can define components T_{V} (M; φ) := _{p}(φ) ⊗_{T_{V}^{0} K} T_{V} M. The assignment (V, φ) ↦ T_{V} (M; φ) defines a functor T_{−}(M; −) : S(K) → K − U. Here the K - module structure on T_{V} (M; φ) is induced from the T_{V} K - module structure on T_{V} M via the algebra map K → T_{V} K, given as composition K ⟶^{γ_{M,V}} H^{∗}V ⊗ T_{V} K ⟶^{ϵ⊗id} _{p} ⊗ T_{V} K ≅ T_{V} K where γ_{M,V} is the map introduced in 1.13 and ϵ is the augmentation on H^{∗}V. The following lemma shows that under suitable assumptions the natural transformation π induces an isomorphism T_{−}(M; −) ≅ T_{−}(M; −) ∘ i ∘ reg.

LEMMA 4.8. Suppose K is noetherian, M is in K_{fg} −U and α : V → V′ is an epimorphism. Then for each φ ∈ Hom_{K} (K, H^{∗}V′) the map α induces an isomorphism

T_{V} (M; α^{∗}φ) → T_{V′}(M; φ).

\underline{Remark:} If G is a compact Lie group, K = H^{∗}BG and M = H_{G}^{∗}X, the mod p - cohomology of the Borel construction of a suitable G space X then this result follows from 5.2 and 5.3 below. In fact, this is the guiding example for the purely algebraic case that we are discussing here.

\underline{Proof:} We may assume that V = V′ ⊕ V′′ and α is the projection. In this case we find (cf. [DW2])

T_{V} (M; α^{∗}φ) ≅ T_{V′′}(T_{V′}(M; φ); ε)

where ε is the trivial algebra map T_{V′}(K; φ) → H^{∗}V′′ adjoint to K →^{φ} H^{∗}V′ →^{α^{∗}} H^{∗}V′⊗H^{∗}V′′. Now T_{V′}(K; φ) is noetherian and connected, and T_{V′}(M; φ) is finitely generated over it by [DW3] or [He], hence we get T_{V′′}(T_{V′}(M; φ); ε) ≅ T_{V′}(M; φ) (cf. [DW1], or [S1] where this is also implicit.).

We are now almost ready to state our main result of this section. But first note that for M ∈ K − U the components T_{V} (M; φ) inherit the H^{∗}V - comodule structure from T_{V} M and the natural transformations μ, ν of 1.16 induce similar natural transformations H^{∗} ⊗ (T_{−}(M; −))^{<n} ⟶ H^{∗} ⊗ (H^{∗} ⊗ (T_{−}(M; −))^{<n} of functors defined on S(K)_{♯} resp. R(K)_{♯}. We will denote these transformations again by μ and ν. Finally the natural transformation λ (cf. 3.4) from the constant functor (defined on E) with value M to H^{∗} ⊗ T_{−}M induces one from the constant functor M (defined on S(K)_{♯} resp. R(K)_{♯}) to H^{∗} ⊗ T_{−}(M; −) which we will again denote by λ. For K = M and (V, φ) →^{α} (V′, φ′) ∈ mor(R(K)) the maps λ(α), μ(α) and ν(α) are all algebra homomorphisms and induce K - module structures on H^{∗}V ⊗ (T_{V′}(M; φ′))^{<n} and H^{∗}V ⊗ (H^{∗}V ⊗ (T_{V′}(M; φ′))^{<n} for each M in K − U. With these K - module structures the transformations λ, μ and ν become natural transformations of functors from S(K)_{♯} resp. R(K)_{♯} to K − U whenever M is in K − U. Furthermore, if M is in K_{fg} − U, then it is easy to see that λ, μ and ν become natural transformations of functors from R(K)_{♯} to K_{fg} − U.

THEOREM 4.9. Suppose K ∈ K is noetherian (of transcendance degree ≤ d) and let M be in K_{fg} − U. Then the canonical map

M → lim_{(V,φ)→^{α}(V′,φ′)}[Eq : H^{∗}V ⊗ (T_{V′}(M; φ′))^{<n} \overrightarrow{⟶}_{ν(α)}^{μ(α)} H^{∗}V ⊗ (H^{∗}V ⊗ (T_{V′}(M; φ′))^{<n}] ,

induced by the maps λ(α) for α ∈ mor(R(K)), is localization away from Nil_{n} and hence by Theorem 4.3 an isomorphism if n is large. (The limit is here taken over the category R(K)_{♯}.)

The remark after 4.5 applies equally well here: because of the finiteness of R(K) this result can be proven without using section 2.

COROLLARY 4.10.

a) If K is noetherian and M ∈ K_{fg} − U, then L_{n}M is in K_{fg} − U.

b) If K is noetherian, then L_{n}K is noetherian.

c) If K is noetherian and M ∈ K_{fg} − U, then L_{n}M ∈ (L_{n}K)_{fg} − U.

\underline{Proof of Theorem 4.9:} Because of 3.4 it suffices to show that we have isomorphisms

lim_{α∈E_{♯}} H^{∗}dα ⊗ (T_{rα}M)^{<n} ≅ lim_{(V,φ)→^{α}(V′,φ′)} H^{∗}V ⊗ (T_{V′}(M; φ′))^{<n}

and

lim_{α∈E_{♯}} H^{∗}dα ⊗ (H^{∗}dα ⊗ T_{rα}M)^{<n} ≅ lim_{(V,φ)→^{α}(V′,φ′)} H^{∗}V ⊗ (H^{∗}V ⊗ T_{V′}(M; φ′))^{<n}

which are compatible with the two morphisms (between the two respective limits) induced by our good old friends, the natural transformations μ and ν.

We will treat the first case and leave the similar second case and the compatibility to the reader. Let E_{∞}^{E} resp. E_{∞}^{S(K)} resp. E_{∞}^{R(K)} denote the category of functors from E resp. S(K) resp. R(K) to E_{∞}. Fix natural numbers j ≤ k ≤ n. We can identify lim_{α∈E_{♯}} H^{j}dα ⊗ T_{rα}^{k−j}M with Hom_{E_{∞}^{E}}(H_{j}, T_{−}^{k−j}).

Next we identify Hom_{E_{∞}^{E}}(H_{j}, T_{−}^{k−j}) with Hom_{E_{∞}^{S(K)}}(H_{j}, T_{−}^{k−j}(M; −)). Now the adjunction between reg and i and the factorization of T_{−}^{k−j}(M; −) given in Lemma 4.8 show that the natural restriction map

Hom_{E_{∞}^{S(K)}}(H_{j}, T_{−}^{k−j}(M; −)) ⟶ Hom_{E_{∞}^{R(K)}}(H_{j}, T_{−}^{k−j}(M; −))

is an isomorphism. Finally we use the canonical isomorphism

Hom_{E_{∞}^{R(K)}}(H_{j}, T_{−}^{k−j}(M; −)) ≅ lim_{(V,φ)→^{α}(V′,φ′)} H^{j}V ⊗ (T_{V′}(M; φ′))^{<n} .

We finish this section by giving a characterization of those M in K − U such that L_{n}M is in K_{fg} − U.

PROPOSITION 4.11. Suppose K is noetherian and M ∈ K − U. Then L_{n}M is in K_{fg} − U iff the following two conditions are satisfied.

(1) For all (V, φ) ∈ R(K) and epimorphisms α : W → V the map T_{W} (M; α^{∗}φ) → T_{V} (M; φ) induced by α is an isomorphism in degrees < n.

(2) (T_{V} (M; φ))^{<n} is finite for all (V, φ) ∈ R(K).

\underline{Proof:} ”⟹” This is a consequence of Lemma 4.8. and the fact that T_{W} M → T_{W} L_{n}M is iso in degrees < n for each W.

”⟸” Assumption (1) guarantees that the localization L_{n}M is given by the expression of 4.9 and condition (2) clearly implies that this expression is in K_{fg} − U.

5. The case of equivariant mod p - cohomology

We now specialize the theory of section 4 to the case of equivariant mod p cohomology.

\underline{5.1.} Let us first fix some notation. Let G be a topological group and X a G - space. As usual EG denotes the total space of the universal G - bundle over BG, the classifying space of G. We are interested in H_{G}^{∗}X := H^{∗}(EG ×_{G} X) := H^{∗}(EG ×_{G} X; _{p}). If G is a compact Lie group and H^{∗}X := H^{∗}(X; _{p}) is finite dimensional as graded vector space then H_{G}^{∗}X is a finitely generated module over the noetherian algebra H^{∗}BG, so we can apply our theory including section 4 to H_{G}^{∗}X.

For a compact Lie group G and a finite G - CW - complex X the functors f^{<n}H_{G}^{∗}X can be described as follows. Let E be an elementary abelian p - subgroup of G, let C_{G}(E) be its centralizer in G and X^{E} ⊂ X the E - fixed points. Then the homomorphism E × C_{G}(E) → C_{G}(E), (e, g) ↦ eg induces a map BE × (EC_{G}(E) ×_{C_{G}(E)} X^{E}) ⟶ EC_{G}(E) ×_{C_{G}(E)} X^{E} which we denote by m_{E}. Now let V be in E and denote by Rep(V, G) the quotient of Hom(V, G) by the conjugation action of G. Represent an element ρ ∈ Rep(V, G) by a homomorphism V → G still denoted by ρ. We abbreviate C_{G}(Imρ) by G_{ρ} and X^{Imρ} by X^{ρ}. The obvious composition

BV × (EG_{ρ} ×_{G_{ρ}} X^{ρ}) ⟶ BImρ × (EG_{ρ} ×_{G_{ρ}} X^{ρ}) ⟶^{m_{Imρ}} (EG_{ρ} ×_{G_{ρ}} X^{ρ}) → EG ×_{G} X

denoted by c_{ρ} induces a map of unstable algebras c_{ρ}^{∗} : H_{G}^{∗}X ⟶ H^{∗}BV ⊗ H_{G_{ρ}}^{∗}X^{ρ} with adjoint ad(c_{ρ}^{∗}) : T_{V} H_{G}^{∗}X ⟶ H_{G_{ρ}}^{∗}X^{ρ}.

THEOREM 5.2 [L3].

a) Let G be a compact Lie group, X a finite G − CW − complex and let V be in E. Then the homomorphism of unstable algebras

T_{V} H_{G}^{∗}X ⟶ ∏_{ρ∈Rep(V,G)} H_{G_{ρ}}^{∗}X^{ρ}

with components ad(c_{ρ}^{∗}) is an isomorphism.

b) The same conclusion holds if X is a point and G is a discrete group of f.v.c.d..

Note that in case G = W is elementary abelian and X is a point this specializes to Proposition 1.12.3.

From now on we assume for simplicity that G is either a compact Lie group or a discrete group of f.v.c.d. such that H^{∗}BG is noetherian. This assumption is satisfied by many interesting f.v.c.d. groups, e.g. general linear groups over the ring of S - integers in a number field [Se] or mapping class groups of orientable surfaces [Ha].

\underline{5.3.} In order to apply our results of section 4 we recall that Rector’s category R(H^{∗}BG) is equivalent to Quillen’s category A(G) whose objects are the elementary abelian p - subgroups of G and whose morphism sets are given by

Hom_{A(G)}(E_{1}, E_{2}) = {α ∈ Hom(E_{1}, E_{2})|∃g ∈ G with αe = geg^{−1} ∀e ∈ E_{1}} .

For the convenience of the reader and for future reference we will briefly explain this equivalence. The canonical map

Rep(V, G) → Hom_{K}(H^{∗}BG, H^{∗}BV), φ ↦ (Bφ)^{∗}

is bijective (see [L1] for example) and can be used to identify the objects in R(H^{∗}BG) with the G - conjugacy classes of injective homomorphisms from elementary abelian p -groups to G. If φ : V → G and φ′ : V′ → G are two such monomorphisms considered as objects of R(H^{∗}BG) then

mor_{R(H^{∗}BG)}(φ, φ′) = {α ∈ Hom(V, V′)|φ′α = φ as elements in Rep(V, G)} . Now choose a skeleton A_{s}(G) of A(G), i.e a full subcategory whose objects are in one to one correspondence to the isomorphism classes of objects in A(G). Then A(G) and A_{s}(G) are equivalent and equivalences A_{s}(G) → R(H^{∗}BG) and R(H^{∗}BG) → A_{s}(G) are given on objects by associating to an elementary abelian p - subgroup E the conjugacy class of the inclusion E ⊂ G and to a conjugacy class of monomorphisms from V to G the unique object in A_{s}(G) isomorphic in A(G) to the images of these monomorphisms.

Furthermore 5.2 implies that if H_{G}^{∗}X is considered as object of H^{∗}BG − U and i = i_{E} denotes the restriction homomorphism H^{∗}BG ⟶ H^{∗}BE then (T_{E}H_{G}^{∗}X)_{i} is isomorphic to H_{C_{G}(E)}^{∗}X^{E} and the H^{∗}E - comodule structure on H_{C_{G}(E)}^{∗}X^{E} is given by m_{E}^{∗}. As before we have natural transformations λ, μ and ν of functors defined on the twisted arrow category (see 1.16) A(G)_{♯}. If n > 0 is fixed and α is in obA(G)_{♯} = morA(G) then

λ(α) : H_{G}^{∗}X ⟶ H^{∗}dα ⊗ (H_{C_{G}(rα)}^{∗}X^{rα})^{<n}

is the composition of c_{rα}^{∗} followed by α^{∗} ⊗ id. ( Here and in the remainder of this section we abbreviate H^{∗}BG by H^{∗}G if G is a finite group.) Furthermore

μ(α) : H^{∗}dα ⊗ (H_{C_{G}(rα)}^{∗}X^{rα})^{<n} ⟶ H^{∗}dα ⊗ (H^{∗}dα ⊗ H_{C_{G}(rα)}^{∗}X^{rα})^{<n}

is given as truncation of ψ_{dα} ⊗ id (cf 1.16) and

ν(α) : H^{∗}dα ⊗ (H_{C_{G}(rα)}^{∗}X^{rα})^{<n} ⟶ H^{∗}dα ⊗ (H^{∗}dα ⊗ H_{C_{G}(rα)}^{∗}X^{rα})^{<n}

is the composition of id ⊗ m_{rα}^{∗} followed by id ⊗ (H^{∗}α ⊗ id)^{<n} (cf.1.16).

Now Theorem 4.9 translates into

THEOREM 5.4.

a) Let G be a compact Lie group and X a finite G − CW − complex. Then the maps λ(α) for α ∈ mor(A(G)) induce a homomorphism of unstable algebras

H_{G}^{∗}X ⟶ lim_{α∈A(G)_{♯}}[Eq : H^{∗}dα ⊗ (H_{C_{G}(rα)}^{∗}X^{rα})^{<n} \overrightarrow{⟶}_{ν(α)}^{μ(α)} H^{∗}dα ⊗ (H^{∗}dα ⊗ H_{C_{G}(rα)}^{∗}X^{rα})^{<n}]

which is localization away from Nil_{n} .

b) The same conclusion holds if X is a point and G is a discrete group of f.v.c.d. such that H^{∗}BG is noetherian.

We recall that for n > d_{0}(H_{G}^{∗}X) this map is a monomorphism and for n > d_{1}(H_{G}^{∗}X) this map is an isomorphism and hence H_{G}^{∗}X is determined by H^{∗}C_{G}(E) in degrees less or equal to d_{1}(H_{G}^{∗}X). More precisely H_{G}^{∗}X is determined by the “truncated comodule functor” (in the sense of 1.14) from A(G) to U^{<n} which sends E ∈ obA(G) to (H_{C_{G}(E)}^{∗}X^{E})^{<n}. The cases n = 1 and n = ∞ in 5.4 deserve some comments. If n = 1 and, say X is a point, then the equalizer of μ(α) and ν(α) is H^{∗}dα and the limit can be identified with Quillen’s lim_{E∈A(G)^{op}} H^{∗}E. If n = ∞ and, say X is a point, the equalizer can be identified with H^{∗}BC_{G}(E) and the inverse limit with lim_{E∈A(G)} H^{∗}BC_{G}(E). Now A(G) has the trivial subgroup {1} as an initial object and hence the last limit agrees with H^{∗}BG in accordance with the fact that the localization λ_{∞} is always an isomorphism.

COROLLARY 5.5. Let G be either a compact Lie group or a discrete group of f.v.c.d. such that H^{∗}BG is noetherian and let n > d_{0}H^{∗}BG. Then the maps λ(id_{E}) for E ∈ A(G) induce a monomorphism

H^{∗}BG ⟶ ∏_{E∈A(G)} H^{∗}BE ⊗ (H^{∗}BC_{G}(E))^{<n} .

COROLLARY 5.6. Let G be as in 5.5 and denote by N(G) the kernel of the first localization

map λ_{1} ,i.e. of the map H^{∗}BG \underrightarrow{∏ res_{E}} ∏_{E∈A(G)} H^{∗}BE. Then the n -th power of N(G) is trivial if n > d_{0}(H^{∗}BG).

\underline{Proof:} If x is in N(G) then x restricts trivially to H^{∗}BE for all E ∈ A(G) and hence λ(id_{E})(x) ∈ H^{∗}BE ⊗ (H̃^{∗}BC_{G}(E))^{<n}. Consequently, if x_{1}, ...x_{n} are in N(G) then λ(id_{E})(x_{1}...x_{n}) = 0 and we are done by 5.5.

\underline{Remark:} If x is in N(G) then x is nilpotent by [Q1]. If p = 2 then H^{∗}BE, E elementary abelian, does not contain nontrivial nilpotent elements, i.e. if x ∈ H^{∗}BG is nilpotent then clearly x is in N(G) and hence N(G) is precisely the radical R(G) of H^{∗}BG. For p odd H^{∗}BE does contain nilpotent elements, of height p at most, and hence R(G) can be larger than N(G); however, if x ∈ H^{∗}BG is nilpotent we have at least x^{p} ∈ N(G) and inspecting the proof of 5.6 gives still x^{n} = 0 if n is the smallest multiple of p larger than d_{0}(H^{∗}BG).

COROLLARY 5.7. Let G be as in 5.5. Assume x ∈ H^{∗}BG is not a zero divisor. Denote by C_{x} the collection of objects E ∈ A(G) such that x restricts non-trivially to H^{∗}BE. Then the map

H^{∗}BG ⟶ ∏_{E∈C_{x}} H^{∗}BC_{G}(E)

whose components are the restriction homomorphisms is a monomorphism.

\underline{Proof:} Take n to be larger than d_{0}H^{∗}BG and apply 5.5. Assume y ∈ H^{∗}BG restricts trivially to H^{∗}BC_{G}(E) for all E ∈ C_{x}. As λ(id_{E}) factors through the restriction from H^{∗}BG to H^{∗}BC_{G}(E) we see that λ(id_{E})(y) = 0 for all E ∈ C_{x}. On the other hand λ(id_{E})(x^{m}) = 0 for all E ∉ C_{x} whenever m is larger than n. Consequently λ(id_{E})(x^{m}y) = 0 for all E and hence x^{m}y = 0 by 5.5. Now x is a non zero divisor and hence y = 0.

\underline{Remark:} Let G be the general linear group GL(n, [\frac{1}{2}]) over the ring [\frac{1}{2}] and let D_{n} be the subgroup of diagonal matrices. Note that D_{n} = C_{G}(E_{n}) where E_{n} is the elementary abelian 2 - subgroup of G consisting of all diagonal matrices of order 2. W.G.Dwyer (private communication) has shown that the restriction homomorphism in mod 2 cohomology H^{∗}(BGL(n, [\frac{1}{2}])) ⟶ H^{∗}(BD_{n}) cannot be injective for all n. Consequently for such n each element in H^{∗}(BGL(n, [\frac{1}{2}])) which restricts nontrivially to H^{∗}BE_{n} and trivially to all other H^{∗}BE, E ∈ A(G), E not conjugate to E_{n}, must be a zero divisor. Now consider the element in H^{2^{n+1}−2}(BGL(n, )) whose restriction to H^{2^{n+1}−2}(BΠ_{i=1}^{n}GL(1, )) is the product of all nonzero elements of degree 2. For n as above this element restricts to a zero divisor in H^{∗}(BGL(n, [\frac{1}{2}])) and hence H^{∗}(BGL(n, [\frac{1}{2}])) is not torsionfree, hence not free as a module over H^{∗}BGL(n, ) with the module structure induced from the embedding of GL(n, [\frac{1}{2}]) into GL(n, ). Thus Quillen’s conjecture in [Q1, p.591] was too optimistic.

COROLLARY 5.8. Let G be a finite group in which all elements of order p are central (and hence form a central elementary abelian p - subgroup C). Then the map λ(id_{C}) induces a homomorphism of unstable algebras

H^{∗}BG ⟶ Eq : H^{∗}BC ⊗ (H^{∗}BG)^{<n} \overrightarrow{⟶}_{ν(id_{C})}^{μ(id_{C})} H^{∗}BC ⊗ (H^{∗}BC ⊗ H^{∗}BG)^{<n}

which is localization away from Nil_{n}.

We remark that in [HP] it is shown that in a suitable sense almost all p - groups satisfy the assumption of Corollary 5.8.

\underline{Proof:} We consider the description of the localization away from Nil_{n} given by 5.4 and rewrite it as

Eq[ lim_{α∈A(G)_{♯}} H^{∗}dα ⊗ (H^{∗}BC_{G}(rα))^{<n} \overrightarrow{⟶}_{ν}^{μ} lim_{α∈A(G)_{♯}} H^{∗}dα ⊗ (H^{∗}dα ⊗ H^{∗}BC_{G}(rα))^{<n}] .

Hence it suffices to show that the two limits are isomorphic to H^{∗}BC ⊗ (H^{∗}BG)^{<n} and H^{∗}BC ⊗ (H^{∗}BC ⊗ H^{∗}BG)^{<n} respectively. To see this observe that the assumptions on G imply that all subgroups E of G are central (hence C_{G}(E) = G for all E) and that C is terminal in A(G). Then we obtain a sequence of isomorphisms

lim_{α} H^{∗}dα ⊗ (H^{∗}BC_{G}(E))^{<n} ≅ lim_{α} H^{∗}dα ⊗ (H^{∗}BG)^{<n} ≅

≅ (lim_{α} H^{∗}dα) ⊗ (H^{∗}BG)^{<n} ≅ H^{∗}BC ⊗ (H^{∗}BG)^{<n} .

For the final isomorphism we have used that for a functor F on A(G)^{op} we always have lim_{A(G)_{♯}} F d ≅ lim_{A(G)^{op}} F (cf. 1.16.1). As the second limit can be treated in the same way we are done.

\underline{5.9.} Theorem 5.4 can be used to describe localizations of H^{∗}BG as in Duflot [D2]. If P ⊂ H^{∗}BG is a minimal prime ideal, given as kernel of the restriction to a maximal elementary abelian p - subgroup E of G, then the localizations (H^{∗}BG)_{P} and (H^{∗}BC_{G}(E))_{P}^{N_{G}(E)} with respect to this prime ideal agree. (Here N_{G}(E) denotes the normalizer of E in G.) This results easily from exactness of localization, the fact that N_{G}(E)/C_{G}(E) is isomorphic to Aut_{A(G)}(E) and that

Eq[H^{∗}dα ⊗ (H^{∗}BC_{G}(rα)^{<n} \overrightarrow{⟶}_{ν(α)}^{μ(α)} H^{∗}dα ⊗ (H^{∗}dα ⊗ H^{∗}BC_{G}(rα))^{<n}]

is trivial after localization unless dα ≅ rα ≅ E (isomorphic in A(G)). With a bit more care one can also recover her localization result for H_{G}^{∗}X.

\underline{5.10.} Next we give an illustration of the discusssion in 3.8 and 3.9, i.e. we discuss the filtration of H^{∗}BG by the k - nilpotent ideals Ker λ_{k}, k = 1, 2, ... which we will also denote by N_{k}(G). It is clear that in this case this discussion can be carried out in A(G)_{♯} instead of E_{♯}. To simplify notation we will write r̃p_{n−1}(G) instead of r̃p_{n−1}H^{∗}BG and we consider this as a functor on A(G). Recall that this functor determines the kernel of the map from L_{n}H^{∗}BG to L_{n−1}H^{∗}BG. Let E be in A(G). Then we get

r̃p_{n−1}(G)(E) = ⋂_{E⟶^{α}E′} Ker[H^{n−1}BC_{G}(E) ⟶^{α_{∗}} H^{n−1}BC_{G}(E′) ⟶ (H̃^{∗}E′⊗H^{∗}BC_{G}(E′))^{n−1}]

where α_{∗} is induced by α and the second arrow is the composition of the map (m_{E′})^{∗} followed by the canonical projection of H^{∗}E′ ⊗ H^{∗}BC_{G}(E′) onto H̃^{∗}E′ ⊗ H^{∗}BC_{G}(E′). By dualizing we see in particular that r̃p_{n−1}(G)(E) = 0 (i.e. N_{n−1}(G) = N_{n}(G)) if and only if the dual map

⨁_{E⟶^{α}E′} H̃_{∗}BE′ ⊗ H_{∗}BC_{G}(E′) ⟶ H_{∗}BC_{G}(E′) ⟶^{α_{∗}} H_{∗}BC_{G}(E)

is onto in degree n−1 for each E ∈ A(G). In particular, if E is a maximal elementary abelian p - subgroup, this means that H_{n−1}BC_{G}(E) is decomposable with respect to the H_{∗}BE - module structure on H_{∗}BC_{G}(E). For n = 2 we get N_{1}(G) = N_{2}(G) if and only if H_{1}BC_{G}(E) is generated by H_{1} of the elementary abelian p -subgroups of C_{G}(E) (for all E ∈ A(G)).

We remark that there are generalizations of the results of 5.5 - 5.10 in the framework of section 4, i.e for K noetherian or more generally for M ∈ K_{fg} − U with K noetherian. In 5.8 we would then have to use an appropriate concept of centers in K_{fg} − U à la Dwyer - Wilkerson [DW2].

\underline{5.11.} Finally we mention that in many cases T_{V} H^{∗}X agrees with H^{∗} map(BV, X) [L2] and then one can describe the localizations for H^{∗}X in terms of the low dimensional cohomology of the mapping spaces map(BV, X) for V ∈ E. If H^{∗}X happens to be noetherian, this description can be simplified along the lines of section 4.

II. The invariants d_{0} and d_{1} in the case of equivariant cohomology; examples

1. The invariants d_{0} and d_{1} in the case of an elementary abelian p - group

In this section we will prove the following result.

THEOREM 1.1. Let V be an elementary abelian p-group and M a smooth compact V - manifold (possibly with boundary). Then d_{1}H_{V}^{∗} M ≤ dim M.

This result fails to be true if we do not restrict to manifolds. This failure is an interesting phenomenon which will be the subject of a future paper. The result we will actually prove is stronger than Theorem 1.1 and in order to formulate it properly we will introduce the following concept.

DEFINITION 1.2. Let C be an unstable A - module. We say that C belongs to the class C_{m,d} if and only if C has an injective resolution

C → I_{0} → I_{1} → I_{2} → ⋯

such that I_{k} is a direct sum of modules of the form H^{∗}W ⊗J(l) with dim W ≤ d and l ≤ m−k.

Note that for C ∈ C_{m,d} we have d_{1}C ≤ m (by I.3.2.(a) and I.3.2.(b) ) and C has finite injective dimension which is at most m.

\underline{1.3.} Next suppose we have an exact sequence of A - modules

0 ⟶ C_{1} ⟶ C_{2} ⟶ C_{3} ⟶ 0 .

Then the following statements are easy to verify \underline{1.3.1.} If C_{1} and C_{3} are in C_{m,d}, then C_{2} is in C_{m,d}. \underline{1.3.2.} If C_{2} is in C_{m,d} and C_{3} is in C_{m−1,d}, then C_{1} is in C_{m,d}. \underline{1.3.3.} It follows easily that if B is an unstable A - module which is trivial in degrees > m, then H^{∗}V ⊗ B is in C_{m,d} if d = dim V.

THEOREM 1.4. Let V be an elementary abelian p-group of rank d and M a smooth V - manifold of dimension m (possibly with boundary). Assume that the subspace of M consisting of all points with isotropy group of rank i has finitely many components for each 0 ≤ i ≤ d. Then H_{V}^{∗} M is in C_{m,d}.

Clearly a smooth compact V - manifold satisfies the assumptions of Theorem 1.4 and hence Theorem 1.1 follows. In fact, the proof will show that in this case the I_{k} in Definition 1.2. can be required to be finite direct sums.

\underline{Proof:} This depends heavily on a result of Duflot ([D1]) which we will briefly recall. Duflot considers the filtration

∅ = M_{<0} ⊂ M_{<1} ⊂ M_{<2}... ⊂ M_{<d+1} = M

where M_{<i} denotes the (open) subset of points in M whose isotropy subgroup have rank < i. This gives rise to a decreasing filtration

H_{V}^{∗} M = F_{0} ⊃ F_{1} ⊃ F_{2} ⊃ ⋯ F_{d} ⊃ F_{d+1} = 0

of H_{V}^{∗} M where F_{i} is defined as F_{i} := Ker H_{V}^{∗} M → H_{V}^{∗} M_{<i}. Now let M_{(i)} denote the subspace of M consisting of those points whose isotropy subgroups have rank = i ; M_{(i)} is a smooth closed V - submanifold of M_{<i+1}. If M_{W,d} denotes the union of components in M_{(i)} whose isotropy subgroup is W (of rank i) and whose codimension in M_{<i+1} is d, then we get an isomorphism

(1.5)  H_{V}^{∗} (M_{<i+1}, M_{<i}) ≅ ⨁_{{dim W =i \atop 0≤d≤m}} H̃_{V}^{∗} (T_{W,d})

where T_{W,d} is the Thom space of the normal bundle ν_{W,d} of M_{W,d} in M_{<i+1}.

The crucial result of Duflot is that the Euler class e ∈ H_{V}^{∗} (M_{W,d}) is a non-zero-divisor and therefore (by the Thom isomorphism) the long exact sequence of the pair (D_{W,d}, S_{W,d}) consisting of the disc- and sphere bundle of ν_{W,d} gives a short exact sequence:

(1.6)  0 ⟶ H̃_{V}^{∗} (T_{W,d}) ⟶ H_{V}^{∗} (M_{W,d}) ⟶ H_{V}^{∗} (S_{W,d}) ⟶ 0 .

In addition she deduces (cf. [D1,p.260]) that the restriction maps induce isomorphisms

(1.7)  F_{i}/F_{i+1} ⟶^{≅} H_{V}^{∗} (M_{<i+1}, M_{<i}) .

With (1.5) - (1.7) at hand we can prove Theorem 1.4 as follows. First by 1.3.1 , (1.5) and (1.7) it suffices to consider H̃_{V}^{∗} (T_{W,d}). Now M_{W,d} which is a smooth V - manifold of dimension m − d and S_{W,d} is a smooth V - manifold of dimension m − 1. Furthermore both M_{W,d} and S_{W,d} satisfy the assumptions of the Theorem and so by induction on m, by (1.6) and by 1.3.2 we are done in case d > 0. If d = 0 we have H^{∗}M_{W,d} = H̃^{∗}T_{W,d}. Because W acts trivially and V/W acts freely on M_{W,d} we get

H_{V}^{∗} M_{W,d} ≅ H^{∗}V ⊗_{H^{∗}V/W} H_{V/W}^{∗} (M_{W,d}) ≅ H^{∗}V ⊗_{H^{∗}V/W} H^{∗}((V/W)\M_{W,d})

and we are done by 1.3.3 above.

Duflot treats only the case p odd in her paper. However, her proof carries over verbatim to the case p = 2 if Chern classes are replaced by Stiefel Whitney classes.

2. The invariants d_{0} and d_{1} in the case of a general compact Lie group

\underline{2.1.} Throughout this section we will consider the following data. U is a compact Lie group, S an elementary abelian p - subgroup of G and T a closed subgroup of U containing S such that H^{∗}BT is (via restriction) a finitely generated free H^{∗}BU - module and H^{∗}BS is (via restriction) a finitely generated free H^{∗}BT - module. Furthermore we assume that one of the following equivalent conditions holds.

a) The homomorphisms π_{0}S → π_{0}T and π_{0}T → π_{0}U are onto (and hence π_{0}T and π_{0}U are p-groups).

b) The natural action of π_{0}T on H^{∗}(T/S) and of π_{0}U on H^{∗}(U/T) is nilpotent.

c) The natural action of π_{0}T on H^{∗}(T/S) and of π_{0}U on H^{∗}(U/T) is trivial.

The implications (a) ⇒ (b) and (c) ⇒ (a) are easy. (For the second one note that triviality of the action in degree 0 imply that T/S and U/T are connected.) For (b) ⇒ (c) one can use the Eilenberg - Moore spectral sequence of the fibrations BS → BT with fibre T/S and BT → BU with fibre U/T to see that for each the cohomology of the total space surjects onto the cohomology of the fibre which implies that the action is trivial.

\underline{2.2 Examples.} a) For any prime p we can take U the unitary group U(n) or the special unitary group SU(n) or the symplectic group Sp(n): in each of these cases T can be taken as a maximal torus of U and S as p - torus, i.e. as the subgroup of T consisting of elements of order p.

b) For any prime p we can take U to be a torus, T = U and S the p - torus of T.

c) For p = 2 we can take U the orthogonal group or the special orthogonal group and T = S a 2 - torus.

d) If (U_{1}, T_{1}, S_{1}) and (U_{2}, T_{2}, S_{2}) satisfy (2.1) then (U_{1} × U_{2}, T_{1} × T_{2}, S_{1} × S_{2}) does, too.

THEOREM 2.3. Let G be a compact Lie group and M a compact smooth G - manifold (possibly with boundary). Assume that the triple (U, T, S) satisfies 2.1.(a) – 2.1.(c) and that G embedds into U. Then

a) d_{0}H_{G}^{∗}M ≤ dim M + dim U/G

b) d_{1}H_{G}^{∗}M ≤ dim M + dim U/G + max{dim U/T, dim T }.

Theorem 2.3 and the examples above show how estimates for d_{0} and d_{1} can be obtained from knowledge of the representation theory of G.

Theorem 2.3 is not immediately applicable to d_{0}H^{∗}BG and d_{1}H^{∗}BG if G is a discrete group of f.v.c.d.. However, for such groups there is often a finite quotient group \overline{G} and a finite \overline{G} - CW - complex X such that H^{∗}BG is isomorphic to H∗ ; \overline{G}X. Important examples are given by general linear groups over rings of S - integers in a number field [Se] and mapping class groups of orientable surfaces [Ha]. Now such an X has the equivariant homotopy type of a compact smooth \overline{G} - manifold M with boundary and Theorem 2.3 applies to H∗ ; \overline{G}M ≅ H^{∗}BG.

Theorem 2.3 will be a consequence of the following result which is essentially taken from [Q1,sect.6].

PROPOSITION 2.4. Assume U is a compact Lie group and T is a closed subgroup such that H^{∗}BT is (via restriction) a finitely generated free H^{∗}BU - module and such that the action of π_{0}U on H^{∗}(U/T) is trivial. Let G be a closed subgroup of U and let X be any G - space. Then the following (coequalizer) diagram of G - spaces

X × (U/T) × (U/T) \overrightarrow{⟶}_{pr_{1}}^{pr_{2}} X × (U/T) ⟶^{pr} X

(in which pr_{1} resp. pr_{2} resp. pr denote the appropriate projection maps and G acts diagonally on the products) induces an equalizer diagram of unstable algebras

H_{G}^{∗}X ⟶^{pr^{∗}} H_{G}^{∗}(X × (U/T)) \overrightarrow{⟶}_{pr_{1}^{∗}}^{pr_{2}^{∗}} H_{G}^{∗}(X × (U/T) × (U/T)) .

\underline{Proof of 2.3:} Consider the action of the group G × S on M × U given by ((g, s), (m, u)) ↦ (gm, gus^{−1}). Both G and S act freely and hence we get isomorphisms

H_{G}^{∗}(M × (U/S)) ≅ H_{G×S}^{∗}(M × U) ≅ H_{S}^{∗}(G\(M × U))

and similarly

H_{G}^{∗}(M × (U/S) × (U/S)) ≅ H_{G×S×S}^{∗}(M × U × U) ≅ H_{S×S}^{∗}(G\(M × U × U))

(with G still acting diagonally.) Furthermore the S × S - action on G\(M × U × U) is still free when restricted to either of the two factors and hence 2.4 gives us an exact sequence

(2.5)  0 ⟶ H_{G}^{∗}M ⟶ H_{S}^{∗}(G\(M × U)) ⟶ H_{S}^{∗}((S × G)\(M × U × U)) .

From this sequence and II.1.1 we get the claimed estimate for d_{0} , i.e.

(2.6)  d_{0}H_{G}^{∗}M ≤ dim M + dim U/G

and also

(2.7)  d_{1}H_{G}^{∗}M ≤ dim M + 2 dim U − dim G .

In order to get the claimed estimate for d_{1} we consider now the action of G × T on M × U given as above by ((g, t), (m, u)) ↦ (gm, gut^{−1}). The same reasoning as above gives us an exact sequence

0 ⟶ H_{G}^{∗}M ⟶ H_{T}^{∗} (G\(M × U)) ⟶ H_{T}^{∗} ((T × G)\(M × U × U))

and hence by I.3.6.b)

d_{1}H_{G}^{∗}M ≤ max{d_{1}H_{T}^{∗} (G\(M × U)), d_{0}H_{T}^{∗} ((T × G)\(M × U × U))} .

Now we apply (2.6) and (2.7) with G and U both replaced by T, and M replaced by G\(M×U) resp. (T × G)\(M × U × U). We obtain

d_{1}H_{T}^{∗} (G\(M × U)) ≤ dim M + dim U/G + dim T
d_{0}H_{T}^{∗} ((T × G)\(M × U × U)) ≤ dim M + 2 dim U − dim T − dim G

and we are done.

The proof shows that there are extensions of 2.3 to noncompact manifolds similar to Theorem II.1.4.

3. The invariants d_{0} and d_{1} in the case of the symmetric construction

\underline{3.1.} Throughout this section the prime p will be 2. We consider the following properties of a class D of unstable modules contained in Nil_{1}:

(3.1.1) If M ∈ D and M′ ⊂ M then M′ ∈ D. (3.1.2) If M_{1} and M_{2} are in D then M_{1} ⊕ M_{2} is in D. (3.1.3) If M ∈ D and R is reduced then M ⊗ R ∈ D.

DEFINITION 3.2. For a class D contained in Nil_{1} we say that an unstable module M belongs to the class C(D) iff

(1) M is reduced and

(2) the cokernel of the localization map λ_{1,M} belongs to D.

Because the cokernel of λ_{1,M} is always in Nil_{1}, the assumption that D is contained in Nil_{1} is no restriction (in connection with Definition 3.2).

The proof of the following result is straightforward.

PROPOSITION 3.3.a) Suppose D satisfies (3.1.1). Let 0 → M → M_{0} → M_{1} be an exact sequence of unstable modules such that M_{0} is Nil - closed and M_{1} is in D. Then M belongs to C(D).

b) Suppose D satisfies (3.1.1). Let 0 → M → M_{0} → M_{1} be an exact sequence of unstable modules such that M_{0} belongs to C(D) and M_{1} is reduced. Then M belongs to C(D).

c) Suppose D satisfies (3.1.2). If M_{1} and M_{2} are in C(D) then M_{1} ⊕ M_{2} is in C(D).

d) Suppose D satisfies (3.1.2) and (3.1.3). If M_{1} and M_{2} are in C(D) then M_{1} ⊗ M_{2} is in C(D).

\underline{3.4.} Each of the following examples of classes D has all three properties (3.1.1) - (3.1.3).

a) D = Nil_{1}. In this case C(D) consists of all reduced modules. (i.e. d_{0} = 0)

b) D = Nil_{n}. In this case C(D) consists of all modules with d_{0}M = 0 and L_{1}M ≅ L_{2}M ≅ ... ≅ L_{n}M.

c) D equal to the class of modules which are both 1 - nilpotent and Nil_{n+1} - reduced. In this case C(D) consists of all modules with d_{0}M = 0, d_{1}M ≤ n. d) D equal to the class of modules which are both n - nilpotent and Nil_{n+1} - reduced (or equivalently are n - fold suspensions of Nil_{1} - reduced modules). In this case C(D) consists of all modules with d_{0}M = 0, d_{1}M ≤ n and L_{1}M ≅ L_{2}M ≅ ... ≅ L_{n}M.

e) D = 0 the class consisting of the trivial module. Then C(D) is just the class of nilclosed modules. (i.e. d_{0} = d_{1} = 0)

Here is the main result of this section. We denote the symmetric group on n -letters by 𝔖_{n} and the symmetric construction E𝔖_{n} ×_{𝔖_{n}} X^{n} on a space X by 𝔖_{n}X.

PROPOSITION 3.5. Suppose D is a class satisfying conditions (3.1.1) - (3.1.3). Let X be a space such that H^{∗}X belongs to C(D). Then H^{∗}𝔖_{n}X belongs to C(D).

The case that H^{∗}X is Nil - closed, which is example 3.4.e) above, was already proved in [GLZ]. The proof in the general case uses the same strategy as in [GLZ].

\underline{Proof of 3.5:} We will give the proof for n = 2. The general case follows then as in [GLZ]. Consider the homomorphisms H^{∗}𝔖_{2}X → H^{∗}X ⊗ H^{∗}X and H^{∗}𝔖_{2}X → H^{∗}B𝔖_{2} ⊗ H^{∗}X induced by the projection q : E𝔖_{2} × X × X → 𝔖_{2}X resp. the Steenrod diagonal Δ : B𝔖_{2} × X → 𝔖_{2}X. The image of q^{∗} are the invariants (H^{∗}X ⊗ H^{∗}X)^{𝔖_{2}} with respect to the action of 𝔖_{2} on H^{∗}X ⊗ H^{∗}X given by permuting the factors. The study of the image of Δ^{∗} leads to the functor R_{1} : U → U of W.M.Singer [Si]. If we identify H^{∗}B𝔖_{2} with _{2}[u] then R_{1}M can be described as the _{2}[u] - submodule of _{2}[u]⊗M generated by the elements St_{1}m := ∑_{i} u^{i} ⊗ Sq^{|m|−i}m for all m ∈ M (cf. [LZ2]). In case M = H^{∗}X this submodule agrees with the image of Δ^{∗}. The functor R_{1} comes with a natural surjection ρ : R_{1}M → ΦM where Φ denotes the “doubling functor”; ρ sends uR_{1}M to 0 and St_{1}m to Φm, the “double of m”. As ΦM is isomorphic to Ĥ^{0}(𝔖_{2}; M ⊗ M) (with 𝔖_{2} again acting by permuting the factors and Ĥ^{0} denoting 0 - th Tate cohomology, i.e. “invariants divided by norms”) we have a natural map (M ⊗ M)^{𝔖_{2}} → ΦM and we get the following diagram

H^{∗}𝔖_{2}X  ⟶^{Δ^{∗}} R_{1}H^{∗}X
↓^{q^{∗}}  ↓^{ρ}
(H^{∗}X ⊗ H^{∗}X)^{𝔖_{2}} ⟶ ΦH^{∗}X

which can be checked to be a pull-back diagram (cf. [Mi], [Z]). This diagram leads to the definition of a functor 𝔖_{2} : U → U which associates to an unstable A - module M the fibre product of (M ⊗ M)^{𝔖_{2}} and R_{1}M over ΦM. Now 3.5 will follow from the next result.

PROPOSITION 3.6. Suppose D is a class satisfying conditions (3.1.1) - (3.1.3). Let M be an unstable A - module which belongs to C(D). Then 𝔖_{2}M belongs to C(D).

\underline{Proof:} Because of 3.3 it suffices to show that R_{1}M belongs to C(D) whenever M is in C(D). This in turn is a consequence of the fact that R_{1}M is Nil - closed whenver M is Nil - closed [GLZ, Lemme 2.1.2.]. In fact, assume M ∈ C(D). Then we have an exact sequence 0 → M → L_{1}M → D → 0 with D ∈ D. Now R_{1} is exact and hence we get an exact sequence 0 → R_{1}M → R_{1}L_{1}M → R_{1}D → 0, hence by definition of R_{1} an exact sequence 0 → R_{1}M → R_{1}L_{1}M → H ⊗ D and we are done.

\underline{3.7.} We close with some remarks. Proposition 3.5 applied to the class D of example 3.4.c) implies the following: if d_{0}M = 0, d_{1}M ≤ n then d_{0}𝔖_{2}M = 0, d_{1}𝔖_{2}M ≤ n. More generally one can show d_{0}𝔖_{2}M ≤ 2d_{0}M, d_{1}𝔖_{2}M ≤ d_{0}M + d_{1}M for each unstable module M. The situation for p odd is more subtle. Although H^{∗}𝔖_{n}X is reduced whenever H^{∗}X is reduced ([Q2]) Proposition 3.5 does not hold for general D. For example H^{∗}𝔖_{n}X need not be Nil - closed if H^{∗}X is Nil - closed [GLZ].

4. Examples I

\underline{4.1.} Let G and H be two compact Lie groups and consider mod p cohomology for any prime p. Then I.3.6 gives

d_{0}H^{∗}B(G × H) = d_{0}H^{∗}BG + d_{0}H^{∗}BH
d_{1}H^{∗}B(G × H) = max{d_{0}H^{∗}BG + d_{1}H^{∗}BH, d_{1}H^{∗}BG + d_{0}H^{∗}BH} .

\underline{4.2.} Let G be a finite abelian group and p be any prime. Because of 4.1 we only have to consider the case that G is of order a power of p. If G = /p then d_{0} = d_{1} = 0 by I.1.6 and (the easy part) of I.1.7. If G = /p^{n} for some n > 1, then d_{0} = 1, d_{1} = 2. To see this note that in this case we have a splitting of A - modules H^{∗}BG ≅ H^{even}BG ⊕ H^{odd}BG into the even and odd part. Furthermore we have H^{odd}BG ≅ ΣH^{even}BG and hence it suffices to show d_{0}H^{even}BG = 0, d_{1}H^{even}BG = 1 which follows easily from the obvious exact sequence 0 → H^{even}BG → H^{∗}B /p → H^{odd}BG → 0. Observe that these values for d_{0}H^{∗}BG and d_{1}H^{∗}BG agree with the upper bounds obtained from II.2.3 (using an embedding of G into U(1).)

\underline{4.3.} If p is any prime then H^{∗}BS^{1} can be identified with the even part of H^{∗}B /p^{n} (n > 1) and hence d_{0}(H^{∗}BS^{1}) = 0, d_{1}(H^{∗}BS^{1}) = 1. If p = 2 then there is an exact sequence 0 → H^{∗}BS^{3} → H^{∗}BS^{1} → Σ^{2}H^{∗}BS^{3} → 0 from which we deduce d_{0}(H^{∗}BS^{3}) = 0, d_{1}(H^{∗}BS^{3}) = 2. Note that in these examples the upper bounds obtainable from II.2.3 are sharp.

If p is odd then H^{∗}BS^{3} is a direct summand of H^{∗}BS^{1} and we find d_{0}(H^{∗}BS^{3}) = 0, d_{1}(H^{∗}BS^{3}) = 1.

\underline{4.4.} If p = 2 and G = D_{8}, the dihedral group of order 8, then d_{0} = d_{1} = 0 by II.3.5. For the dihedral groups D_{2^{n}} of order 2^{n} we get the same invariants as their mod 2 cohomology happens to be isomorphic (in K) to the one of D_{8} (cf. [MiP]).

\underline{4.5.} If p = 2 and G = 𝔖_{n}, the symmetric group on n letters, then again d_{0} = d_{1} = 0 by II.3.5.

\underline{4.6.} Let p = 2 and G be a quaternion group Q_{2^{n}} of order 2^{n}. Then G embedds into S^{3} and hence II.2.3 implies d_{0} ≤ 3, d_{1} ≤ 5. If we look at the known computation of H^{∗}BG (cf. [MiP]), which can be interpreted as giving an isomorphism of unstable algebras H^{∗}BG ≅ H^{∗}BS^{3} ⊗ H^{∗}(S^{3}/G), and if we apply the tensor product formula I.3.6 (together with 4.3 above) we see that these estimates are sharp. These invariants can also easily be read off from the 2 - local stable splitting

BQ_{2^{n}} ≃ BSL_{2} _{q} ∨ Σ^{−1}(BS^{3}/BN) ∨ Σ^{−1}(BS^{3}/BN)

described in [MiP]. (Here _{q} is a finite field of odd order q such that Q_{2^{n}} is a 2 - Sylow subgroup of SL_{2} _{q}, i.e. q^{2} − 1 = 2^{n}q′ with q′ odd, and N is the normalizer of a maximal torus in S^{3}.) If u_{4} is the periodicity operator in H^{4}BG then H^{∗}BSL_{2} _{q} ≅ _{2}[u_{4}] ⊕ Σ^{3} _{2}[u_{4}] and H^{∗}Σ^{−1}(BS^{3}/BN) ≅ _{2}[u_{4}] ⊗ J(2) (each time this is an isomorphism in U) and hence we get

d_{0}(H^{∗}BSL_{2} _{q}) = 3, d_{1}(H^{∗}BSL_{2} _{q}) = 5
d_{0}(H^{∗}Σ^{−1}(BS^{3}/BN)) = 2, d_{1}(H^{∗}Σ^{−1}(BS^{3}/BN)) = 4 .

\underline{4.7.} Let p = 2 and G = SD_{2^{n}} the semidihedral group of order 2^{n} with n > 3, i.e. G = ⟨x, y|x^{2^{n−1}} = y^{2} = 1, yxy^{−1} = x^{2^{n−2}−1}⟩. Now G embedds into U(2) (via the induced representation of a faithful representation of the cyclic subgroup generated by x) and hence II.2.3 gives the upper bounds d_{0} ≤ 4, d_{1} ≤ 6. On the other hand if _{q} is a finite field with q ≡ 3 mod 4 and q^{2} − 1 = 2^{n−1}q′ with q′ odd then G is a 2 - Sylow subgroup of GL_{2} _{q} and Martino [Ma] (see also [MaP]) showed that there is a 2 - local stable splitting

BSD_{2^{n}} ≃ BGL_{2} _{q} ∨ Σ^{−1}(BS^{3}/BN) .

We will see in II.5.4 below that

d_{0}(H^{∗}BGL_{2} _{q}) = 0, d_{1}(H^{∗}BGL_{2} _{q}) = 2 ,

and hence 4.6 implies

d_{0}(H^{∗}BSD_{2^{n}}) = 2, d_{1}(H^{∗}BSD_{2^{n}}) = 4 .

5. Examples II; the localizations L_{1}, L_{2} and L_{3} for some general linear groups

In this section we will illustrate our theory in the case of the mod p cohomology of the groups GL(n, Λ) for various rings Λ which will be taken from the following list consisting of: the real numbers , the complex numbers , the finite fields _{q} or the ring of S - integers in an algebraic number field. In all these cases Theorem I.5.4 holds.

\underline{5.1.} Fix a prime p. If we denote the Quillen category of GL(n, Λ)) by A(n, Λ) and Rector’s category of H^{∗}BGL(n, Λ) by R(n, Λ), then we get isomorphisms

L_{1}H^{∗}BGL(n, Λ) ≅ lim_{E∈A(n,Λ)} H^{∗}BE ≅ lim_{(V,φ)∈R(n,Λ)} H^{∗}BV .

\underline{5.2.} From now on we will always assume that Λ satisfies in addition the following assumptions:

a) p is invertible in Λ.

b) Λ contains all p - th roots of unity.

c) Projective modules over Λ are free.

In this case the categories R(n, Λ) are very simple. By (a) – (c) every representation of an elementary abelian p -group V on Λ^{n} is isomorphic to a direct sum of one dimensional representations. After choosing an embedding of /p into the group of units Λ^{×} and using I.5.3 we can identify the objects in R(n, Λ), i.e. the faithful representations of elementary abelian p - groups V, with formal sums Σ_{χ}n_{χ}χ with χ running through the characters of V, the n_{χ} being nonnegative integers such that Σ_{χ}n_{χ} = n and such that the set of χ with n_{χ} > 0 spans V^{∗}, the group of characters of V. Furthermore, if φ = Σ_{χ}n_{χ}χ (resp. φ′ = Σ_{χ}n′_{χ}χ) are faithful representations of V (resp. W) then Hom_{R(n,Λ)}(φ, φ′) = {α ∈ Hom(V, W)|φ = Σ_{χ}n′_{χ}χα}. In particular we see that the categories R(n, Λ) are independent of Λ. We will denote them by R_{n}.

Now choose n - linearly independent characters χ_{1}, χ_{2}, ..., χ_{n} of an n - dimensional _{p} - vector space V_{n} and consider the representation φ_{0} := χ_{1} + χ_{2} + ... + χ_{n} as element in R_{n}. Then it is easy to see that for each φ ∈ R_{n} the set Hom_{R_{n}}(φ, φ_{0}) is non-empty and the action of the group Aut_{R_{n}}(φ_{0}), i.e. of the symmetric group 𝔖_{n} on n letters, on this set is transitive. This implies that L_{1}H^{∗}BGL(n, Λ) can be identified with the invariants (H^{∗}BV_{n})^{𝔖_{n}} or with (H^{∗}BE_{n})^{𝔖_{n}} where E_{n} is the subgroup of all diagonal matrices in GL(n, Λ) whose diagonal entries are p -th roots of unity.

\underline{5.3.} Next we consider L_{2}H^{∗}BGL(n, Λ). For simplicity we specialize to the case p = 2 and take Λ to be either the real numbers , the complex numbers , a finite field _{q} of odd order or the ring [\frac{1}{2}]. In all these cases conditions (a) – (c) of 5.2 above are satisfied.

We abbreviate H^{∗}BGL(n, Λ) by M(n, Λ) or by M(n) if Λ is irrelevant or clear from the context. Our analysis will use the sequence

0 ⟶ Σmk_{2}M(n, Λ) ⟶ L_{2}M(n, Λ) ⟶^{τ_{1}} L_{1}M(n, Λ) ⟶ Σmc_{1}M(n, Λ)

of I.3.8 and for this we will determine the exact sequence of functors in F^{<2}

0 ⟶ Σk_{2}M(n, Λ) ⟶ f^{<2}L_{2}M(n, Λ) ⟶^{f^{<2}τ_{1}} f^{<2}L_{1}M(n, Λ) ⟶ Σc_{2}M(n, Λ) ⟶ 0 .

In fact, by I.4 it is clear that we can work with the corresponding functors on the category R_{n} (instead of E) for which we will still use the same notation.

\underline{5.3.1.} We will list the values of these functors in the table below. For this we will introduce the following notation: F_{n} will denote the functor f^{<2}L_{1}M(n, Λ) (considered as functor on R_{n}); by 5.2 above this functor is independent of Λ. The constant functor on R_{n} with value _{2} will be denoted by \underline{_{2}} ; this is also the subfunctor of F_{n} consisting of the homogeneous elements of degree 0. The quotient of F_{n} by this subfunctor is a suspension of a functor which we will call F̃_{n}.

Λ  Σk_{2}M(n)  f^{<2}L_{2}M(n)  f^{<2}L_{1}M(n)  Σc_{2}M(n)
0  F_{n}  F_{n}  0
0  \underline{_{2}}  F_{n}  ΣF̃_{n}
q, q ≡ 3 mod 4  0  F_{n}  F_{n}  0
q, q ≡ 1 mod 4  ΣF̃_{n}  \underline{_{2}} ⊕ΣF̃_{n}  F_{n}  ΣF̃_{n}
[\frac{1}{2}]  ΣF̃_{n}  F_{n} ⊕ ΣF̃_{n}  F_{n}  0

We pause to explain this table:

a) The entries for Λ = are obtained from the a priori knowledge that H^{∗}BGL(n, ) is Nil - closed and hence equal to all its localizations.

b) The functors f^{<2}L_{2}M(n, Λ) are determined by I.5.2: if φ = Σ_{χ}n_{χ}χ is in R_{n} then the centralizer G_{φ} is isomorphic to the product ∏_{χ} GL(n_{χ}, Λ) and hence we find

f^{<2}M(n, Λ)(Σ_{χ}n_{χ}χ) ≅ (⨂_{χ} H^{∗}BGL(n_{χ}, Λ))^{<2} .

c) Now (b) together with F_{n} = f^{<2}L_{2}H^{∗}BGL(n, ) implies that

F_{n}(Σ_{χ}n_{χ}χ) ≅ (⨂_{χ}(H^{∗}E_{n_{χ}})^{𝔖_{nχ}})^{<2}

where E_{n_{χ}} is the appropriate diagonal elemenentary abelian 2 - subgroup of GL(n_{χ}, Λ) as in 5.1 above. d) The entries for Λ = are obtained from the vanishing of H^{1}BGL(n, ) for all n.

e) In all cases, the determinant induces an isomorphism between the mod 2 homology H_{1}(BGL(m, Λ); _{2}) and Λ^{×} ⊗ _{2}. Hence we get H^{1}BGL(m, _{q}) ≅ _{2} and H^{1}BGL(m, [\frac{1}{2}]) ≅ _{2} ⊕ _{2}.

f) Then one shows that the map f^{<2}τ_{1}(Σ_{χ}n_{χ}χ) above identifies with the restriction homomorphism from (⨂_{χ} H^{∗}BGL(n_{χ}, Λ))^{<2} to (⨂_{χ}(H^{∗}E_{n_{χ}})^{𝔖_{nχ}})^{<2} and hence is an isomorphism if Λ = _{q} with q ≡ 3 mod 4, trivial if Λ = _{q} with q ≡ 1 mod 4 and an epimorphism if Λ = [\frac{1}{2}]. It follows that the remaining values of Σk_{2}M(n) and Σc_{2}M(n) are as claimed. We leave it now to the reader to verify that the remaining values of f^{<2}L_{2}M(n) are also as stated. Note that the direct sum decompositions not only respect the A - module structure but also the “comodule structure” on these functors.

\underline{5.3.2.} We already know that m^{<2}(F_{n}) ≅ (H^{∗}BV_{n})^{𝔖_{n}}. So in order to describe L_{2}H^{∗}BGL(n, Λ) we need to understand the unstable A - modules

m^{<2}(ΣF̃_{n}) ≅ Σm(F̃_{n}) ≅ Σ lim_{R_{n}_{♯}} F̃_{n}r ⊗ H^{∗}d ≅ Σ Hom_{E^{R{op \atop n}}} ((F̃_{n})^{∗}, H^{∗})

where (F̃_{n})^{∗} is the contravariant functor whose value on φ ∈ R_{n} is (F̃_{n}(φ))^{∗}. Of course, H^{∗} is the functor whose value on φ : K ⟶ H^{∗}V is H^{∗}V. The group Aut_{R_{n}}(φ_{0}) ≅ 𝔖_{n} acts on (F̃_{n})^{∗}(φ_{o}) ≅ ( _{2})^{n} by permuting the summands, i.e. (F̃_{n})^{∗}(φ_{0}) ≅ _{2}[𝔖_{n−1}\𝔖_{n}] as right _{2}[𝔖_{n}] - module. Furthermore it is easy to see that the natural map

p Hom_{R_{n}}(φ, φ_{0}) ⨂_{_{p}Aut_{Rn} (φ_{0})} (F̃_{n})^{∗}(φ_{0}) ⟶ (F̃_{n})^{∗}(φ)

is an isomorphism for each φ ∈ R_{n}. Consequently we get

m(F̃_{n}) ≅ Hom_{_{2}[𝔖_{n}]} ( _{2}[𝔖_{n−1}\𝔖_{n}], H^{∗}V_{n}) ≅ (H^{∗}V_{n})^{𝔖_{n−1}} .

For Λ = left exactness of m^{<2} yields an exact sequence

0 ⟶ L_{2}H^{∗}BGL(n, ) ⟶ (H^{∗}BV_{n})^{𝔖_{n}} ⟶ Σ(H^{∗}BV_{n})^{𝔖_{n−1}} .

It is not hard to check that this identifies L_{2}H^{∗}BGL(n, ) with the squares in (H^{∗}BV_{n})^{𝔖_{n}}, and hence one can conclude that H^{∗}BGL(n, ) is Nil_{2} - closed. (Of course, this can also be seen more directly!) The computation of L_{2}H^{∗}BGL(n, Λ) for the remaining values of Λ is now immediate.

\underline{5.4.} In the remainder of this section we will outline how our methods can be used to recover Quillen’s computation of the mod 2 - cohomology of GL(n, _{q}) if q ≡ 3 mod 4. We denote the elementary abelian 2 - subgroup of GL(n, _{q}) consisting of all diagonal matrices with diagonal entries ±1 as above by E_{n}. Then H^{∗}BE_{n} is a polynomial algebra _{2}[x_{1}, ..., x_{n}]. Let w_{1}, ..., w_{n} be the elementary symmetric polynomials in the x_{i}. In [Q3] Quillen proved

THEOREM 5.4.1. The restriction homomorphism res : H^{∗}BGL(n, _{q}) ⟶ H^{∗}BE_{n} maps H^{∗}BGL(n, _{q}) isomorphically onto the subalgebra of H^{∗}BE_{n} generated by the elements w_{i}^{2} and Sq^{i−1}w_{i}.

\underline{5.4.2.} As in [Q3] the input in our approach is the knowledge of H^{∗}BGL(n, _{q}) for n ≤ 2. For n = 1 we have trivially H^{∗}BGL(1, _{q}) ≅ H^{∗}BE_{1}.

For n = 2 the situation is more subtle. For the convenience of the reader we give a brief outline of a proof which is somewhat different from Quillen’s.

The mod 2-cohomology of SL(2, _{q}) is well known to be isomorphic to _{2}[u_{4}] ⊗ E(v_{3}) where u_{4} is a polynomial generator of degree 4 and E(v_{3}) is an exterior algebra on a generator of degree 3.

Consider the spectral sequence of the extension SL(2, _{q}) → GL(2, _{q}) → ( _{q})^{×}. We claim that this spectral sequence collapses at its E_{2} -term. Indeed, because the extension is split exact, there are no non-trivial differentials which end on the base; in particular, v_{3} is a permanent cycle. The element u_{4} is also a permanent cycle; it is the restriction of the second Chern class c_{2} of the complex representation of GL(2, _{q}) which is induced from a faithful 2-dimensional representation ρ of a 2-Sylow subgroup S of GL(2, _{q}). (Note that S is a semidihedral group and ρ can be taken as in II.4.7.)

Now we consider the restriction homomorphism res : H^{∗}BGL(2; _{q}) ⟶ H^{∗}BE_{2}. It is straightforward to check that c_{2} restricts to w_{2}^{2} and the generator coming from H^{1} of the base (let us call it e_{1}) to w_{1}. The element Sq^{1}w_{2} is a bit more subtle; by a counting argument it is hit if and only if res : H^{∗}BGL(2, _{q}) ⟶ H^{∗}BE_{2} is injective. The injectivity of res follows from I.5.7; the spectral sequence computation shows that c_{2}e_{1} is not a zero divisor, clearly c_{2}e_{1} restricts trivially to all nonmaximal elementary abelian p subgroups and furthermore we have C_{G}(E_{2}) = E_{2}.

\underline{5.4.3.} We reinterpret the result for n = 2 as follows.

The restriction homomorphism res maps H^{∗}BGL(2, _{q}) to the invariants (H^{∗}BE_{2})^{𝔖_{2}}. In fact, this map is the Nil_{i} - localization for i = 1, 2 by 5.2 and 5.3 above. We want to describe the image of res differently.

For this let C be the central /2 in GL(2, _{q}) consisting of the matrices ±Id. The image of the restriction from (H^{∗}BE_{2})^{𝔖_{2}} to H := H^{∗}BC consists of the squares, i.e. of ΦH and by 5.4.2 the image of res consists exactly of the elements in (H^{∗}BE_{2})^{𝔖_{2}} which further restrict to 4-th powers in H, i.e. of Φ^{2}H. Now the quotient of ΦH by Φ^{2}H can be identified with Σ^{2}ΦH which embedds into Σ^{2}H and we conclude that there is an exact sequence

0 ⟶ H^{∗}BGL(2, _{q}) ⟶ (H^{∗}BE_{2})^{𝔖_{2}} ⟶^{δ} Σ^{2}H

in which the second arrow is induced by res and δ is the composition of the two maps (H^{∗}BE_{2})^{𝔖_{2}} ⟶ ΦH ⟶ Σ^{2}H. In particular we see that the invariants d_{0} and d_{1} take the values 0 and 2 in the case of H^{∗}BGL(2, _{q}). In fact, H^{∗}BGL(2, _{q}) belongs to the class C(D) of II.3.4.d.

\underline{5.4.4.} Now we use Quillen’s argument that a 2-Sylow subgroup S of GL(n, _{q}) is contained in the wreath product GL(2, _{q})≀𝔖_{m} if n = 2m resp. in (GL(2, _{q})≀𝔖_{m})×( _{q})^{×} if n = 2m+1. Then Proposition II.3.5 and 5.4.3 imply d_{0}H^{∗}BGL(n, _{q}) = 0 and d_{1}H^{∗}BGL(n, _{q}) ≤ 2. In particular the localization away from Nil_{3} is an isomorphism.

\underline{5.4.5.} Finally we will describe H^{∗}BGL(n, _{q}) ≅ L_{3}H^{∗}BGL(n, _{q}) for n > 2. As in 5.3 we abbreviate H^{∗}BGL(n, _{q}) by M(n) and we use the sequence

0 ⟶ Σ^{2}mk_{3}M(n) ⟶ L_{3}M(n) ⟶ L_{2}M(n) ⟶ Σ^{2}mc_{3}M(n)

of I.3.8 and the exact sequence of functors

0 ⟶ Σ^{2}k_{3}M(n) ⟶ f^{<3}L_{3}M(n) ⟶^{f^{<3}τ_{2}} f^{<3}L_{2}M(n) ⟶ Σ^{2}c_{3}M(n) ⟶ 0 .

Again we will consider this as an exact sequence of functors defined on R_{n}.

As in 5.3.1 the map f^{<3}τ_{2}(Σ_{χ}n_{χ}χ) is given by the restriction homomorphim from (⨂_{χ} H^{∗}BGL(n_{χ}, Λ))^{<3} to (⨂_{χ}(H^{∗}E_{n_{χ}})^{𝔖_{nχ}})^{<3}. In order to evaluate this we need to know H^{2}BGL(m, _{q}). We already know that d_{0} = 0 and hence H^{∗}BGL(m, _{q}) embedds via restriction into (H^{∗}BE_{m})^{𝔖_{m}}. In particular we get k_{3}M(n) = 0. The class w_{1} is in the image of res and hence w_{1}^{2} is as well. So it remains to consider w_{2}. However, the computation in case GL(2, _{q}) showed that w_{2} is not in the image of res for m = 2 and hence the same must be true for all m ≥ 2. Consequently we get c_{3}M(n)(Σ_{χ}n_{χ}χ) ≅ _{2}^{⊕r} if r is the number of χ with n_{χ} > 1.

Now consider m(c_{3}M(n)) ≅ Hom_{E^{R{op \atop n}}} ((c_{3}M(n))^{∗}, H^{∗}) where (c_{3}M(n))^{∗} is the functor defined via (c_{3}M(n))^{∗}(φ) = (c_{3}M(n)(φ))^{∗}. This functor can be described as follows. Let V_{n−1} = ( /2)^{n−1} and choose a dual basis χ_{1}, χ_{2}, ... χ_{n−1} of V_{n−1}^{∗}. Consider the object φ_{1} := 2χ_{1} + Σ_{i=2}^{n−1}χ_{i} of R_{n}. Then (c_{3}M(n))^{∗}(φ_{1}) ≅ _{2} and the natural map

2 Hom_{R_{n}}(φ, φ_{1}) ⨂_{_{2}Aut_{Rn} (φ_{1})} (c_{3}M(n))^{∗}(φ_{1}) ⟶ (c_{3}M(n))^{∗}(φ)

is an isomorphism for each φ ∈ R_{n}. Consequently we get

m(c_{3}M(n)) ≅ Hom_{_{2}Aut_{Rn} (φ_{1})} ( _{2}, H^{∗}) ≅ (H^{∗}V_{n−1})^{Aut_{Rn} (φ_{1})} .

The group Aut_{R_{n}}(φ_{1}) identifies with Σ_{n−2} acting on V_{n−1} by permuting the last n − 2 summands. Using left exactness of m^{<3} we arrive at an exact sequence

0 ⟶ H^{∗}BGL(n, _{q}) ⟶ (H^{∗}BV_{n})^{𝔖_{n}} ⟶ Σ^{2}H ⊗ (H^{∗}V_{n−2})^{𝔖_{n−2}} .

Then one checks that the last map in this exact sequence is the composition of the inclusion from (H^{∗}BV_{n})^{𝔖_{n}} to (H^{∗}BV_{2})^{𝔖_{2}} ⊗ (H^{∗}BV_{n−2})^{𝔖_{n−2}} with δ ⊗ id_{(H^{∗}BV_{n−2})^{𝔖_{n−2}}} (where δ is as in 5.4.3 above). Finally it is an algebraic exercise to identify the kernel of the map (H^{∗}BV_{n})^{𝔖_{n}} ⟶ Σ^{2}H ⊗ (H^{∗}V_{n−2})^{𝔖_{n−2}} with the subalgebra described in Theorem 5.4.1.

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[Q2] D. Quillen, The Adams Conjecture, Topology 10 (1971), 67-80

[Q3] D. Quillen, On the cohomology and K - theory of the general linear groups over a finite field , Ann. of Math. 96 (1972), 552-586

[Q4] D. Quillen, Higher Algebraic K-theory I, in “Algebraic K-Theory I”, Lect. Notes in Math. 341, Springer 1973, 85-147

[R] D. Rector, Noetherian cohomology rings and finite loop spaces with torsion, J. P. A. A. 32 (1984), 191-217 [S1] L. Schwartz, La filtration nilpotente de la catégorie U et la cohomologie des espaces

de lacets, in “Algebraic Topology - Rational Homotopy” , Lect. Notes in Math. 1318, Springer 1988, 208-218

[S2] L. Schwartz, Notes of a course given at the University of Chicago

[Se] J.P. Serre, Arithmetic groups, in “Homological Group Theory”, London Math. Soc. LNS 36, Cambridge University Press (1979), 105-136

[Si] W.M. Singer, The construction of certain algebras over the Steenrod algebra, J. P. A. A. 11 (1977), 53-59

[Z] S. Zarati, Défaut de stabilité d’opération cohomologiques, Publ. Math. d’Orsay 78-07, 1978

Hans–Werner Henn  Jean Lannes  Lionel Schwartz
Mathematisches Institut  Centre de Mathématiques  Université de Paris-Nord
der Universität  de l´ Ecole Polytechnique  Institut Galilée
Im Neuenheimer Feld 288  Plateau de Palaiseau  Mathématiques
D–6900 Heidelberg  F-91128 Palaiseau Cedex  UA 742 du CNRS
Fed. Rep. of Germany  France  F-93430 Villetaneuse

France