ON UNIVERSALLY STABLE ELEMENTS

Ian J. Leary, Björn Schuster, and Nobuaki Yagita

Abstract. We show that certain subrings of the cohomology of a finite p-group P may be realised as the images of restriction from suitable virtually free groups. We deduce that the cohomology of P is a finite module for any such subring. Examples include the ring of ‘universally stable elements’ defined by Evens and Priddy, and rings of invariants such as the mod-2 Dickson algebras.

Let P be a finite p-group, and let C_{u} be the category whose objects are the subgroups of P, with morphisms all injective group homomorphisms. Let C be any subcategory of C_{u} such that P is an object of C, and such that for any object Q of C, the inclusion of Q in P is a morphism in C. Let H^{∗}( ·) stand for mod-p group cohomology, which may be viewed as a contravariant functor from C_{u} to F_{p}-algebras. We shall study the limit I(P, C) of this functor:

I(P, C) = lim_{Q∈C} H^{∗}(Q).

Given our assumptions on C, we may identify I(P, C) with a subring of H^{∗}(P). In the final remarks we discuss generalizations of our results in which most of the conditions that we impose upon P, C, and H^{∗}( ·) are weakened.

The classical case of this construction occurs in Cartan-Eilenberg’s description of the image of the cohomology of a finite group in the cohomology of its Sylow subgroup as the ‘stable elements’ [2]. Let G be a finite group with P as its Sylow subgroup, and let C_{G} be the subcategory of C_{u} containing all the objects, but with morphisms only those homomorphisms Q to Q′ induced by conjugation by some element of G. Then the image, Im(Res_{P}^{G}), of H^{∗}(G) in H^{∗}(P) is equal to I(P, C_{G}).

Rings of invariants also arise in this way. If C is a category whose only object is P, with morphisms a subgroup H of Aut(P), then I(P, C) is just the subring H^{∗}(P)^{H} of invariants under the action of H.

Another case considered already, which motivated our work, is the ring of universally stable elements defined by Evens-Priddy in [4]. Let C_{s} be the subcategory of C_{u} generated by all subcategories of the form C_{G} as defined above. Then I(P, C_{s}) is the subring I(P) of H^{∗}(P) introduced in [4], consisting of those elements of H^{∗}(P) which are in the image of Res_{P}^{G} for every finite group G with Sylow subgroup P.

A fourth case of interest is I(P, C_{u}), which might be viewed as the elements of H^{∗}(P) which are ‘even more stable’ than the elements of I(P, C_{s}). It is easy to see that in general C_{s} is strictly contained in C_{u}. For example, the endomorphism monoid Hom_{C_{s}}(P, P) of P is the subgroup of Aut(P) generated by elements of order coprime to p, whereas Hom_{C_{u}}(P, P) is the whole of Aut(P). Our main result is the following theorem.

Theorem 1. Let P be a finite p-group, and let C be any subcategory of C_{u} = C_{u}(P) satisfying the conditions stated in the first paragraph. Then there exists a discrete group Γ containing P as a subgroup such that:

a) Im(Res_{P}^{Γ} ) is equal to I(P, C);

b) (Ker(Res_{P}^{Γ} ))^{2} is trivial;

c) Res_{P}^{Γ} induces an isomorphism from H^{∗}(Γ)/\sqrt{0} to I(P, C)/\sqrt{0};

d) Γ is virtually free. More precisely, Γ has a free normal subgroup of index dividing |P |!.

If Γ′ is a free normal subgroup of Γ of finite index, then P maps injectively to the finite group Γ/Γ′, so by the Evens-Venkov theorem [5], H^{∗}(P) is a finite module for H^{∗}(Γ/Γ′) and hence a fortiori for H^{∗}(Γ). Thus one obtains the following corollary.

Corollary 2. Let P and C be as in the statement of Theorem 1. Then H^{∗}(P) is a finite module for its subring I(P, C).

The case C = C_{s} is Theorem A of [4]. Our result is stronger, since it applies to categories such as C_{u} itself, and our proof is more elementary. There is an even shorter proof of Corollary 2 however, which is to deduce it from the following simpler theorem.

Theorem 3. Let P be a finite p-group, and let G be the symmetric group on a set X bijective with P. Regard P as a subgroup of G via a Cayley embedding (or regular permutation representation). Then Im(Res_{P}^{G}) is contained in I(P, C_{u}).

To deduce Corollary 2 from Theorem 3, note that for any C as above, one has

Im(Res_{P}^{G}) ⊆ I(P, C_{u}) ⊆ I(P, C) ⊆ H^{∗}(P),

and H^{∗}(P) is a finite module for Im(Res_{P}^{G}) by the Evens-Venkov theorem. Proof of Theorem 3. We deduce Theorem 3 from the following group-theoretic lemma.

Lemma 4. Let Q ≤ P ≤ G be as in the statement of Theorem 3, and let ϕ be any injective homomorphism from Q to P. Then there exists g ∈ G such that for all q ∈ Q, ϕ(q) = gqg^{−1}.

Proof. Fix a bijection between P and the set X permuted by G. This fixes an embedding i_{P} of P in G. Let i_{Q} be the induced inclusion of Q in G. Write ^{i_{P}}X for X viewed as a P -set. Thus ^{i_{P}}X is a free P -set of rank one. There are two ways to view X as a Q-set, either via i_{Q} or i_{P} ∘ ϕ. The Q-sets ^{i_{Q}}X and ^{i_{P} ∘ϕ}X are both free of rank equal to the index, |P : Q|, of Q in P. Let g be an isomorphism of Q-sets between ^{i_{Q}}X and ^{i_{P} ∘ϕ}X. Then g is an element of G having the required property, because for each x ∈ X and q ∈ Q, g · q · x = ϕ(q) · g · x.

Returning now to the proof of Theorem 3, any morphism in C_{u} factors as the composite of an isomorphism followed by an inclusion. Thus it suffices to show that for ϕ as in Lemma 4, Res_{Q}^{G} and ϕ^{∗} ∘ Res_{P}^{G} are equal. Writing c_{g} for the automorphism of G given by conjugation by g, we have shown that there exists g such that c_{g} ∘ i_{Q} = i_{P} ∘ ϕ. But c_{g}^{∗} is the identity map on H^{∗}(G), and hence i_{Q}^{∗} = ϕ^{∗} ∘ i_{P}^{∗} as required.

This completes the proofs of all of our statements except for Theorem 1. For the proof of Theorem 1 we recall the following theorem (see for example [3], I.7.4 or IV.1.6).

Theorem 5. Let Γ be a group that acts simplicially (i.e., without reversing any edges) on a tree with all stabilizer groups of order dividing a fixed integer M. Then there is a homomorphism from Γ to the symmetric group on M letters whose kernel, K, is torsionfree. Since K acts freely, simplicially on the tree, it follows that K is a free group.

In fact, the short direct proof of Theorem 3 is based on some of the ideas in the proof of Theorem 5 given in [3]. Proof of Theorem 1. The group Γ will be constructed as the fundamental group of a graph of groups (see [3], I.3, [6], I.5, or [1], VII.9 for the definitions and basic theorems). Let Q_{1}, …, Q_{M} be the objects of C, and let ϕ_{1}, …, ϕ_{N} be the morphisms of C. Define a function m so that the domain of ϕ_{i} is Q_{m(i)}. Now let Γ be the group generated by the elements of P and new elements t_{1}, …, t_{N} subject to all relations that hold in P, together with the relations

t_{i}qt_{i}^{−1} = ϕ_{i}(q),

for all i ∈ {1, …, N} and all q ∈ Q_{m(i)}. Thus Γ is the fundamental group of a graph of groups with one vertex and N edges. The vertex group is of course P and the ith edge group is Q_{m(i)}. The two maps from the ith edge group to the vertex group (corresponding to its initial and terminal ends) are the inclusion and ϕ_{i}.

The group Γ as defined above has the following properties (see any of the references listed above): P is a subgroup of Γ; for each i, the homomorphism ϕ_{i}: Q_{m(i)} → P is inner in Γ (i.e., is induced by conjugation by the element t_{i}); Γ acts simplicially on a tree T, with one orbit of vertices and N orbits of edges, with P being a vertex stabilizer and Q_{m(i)} being the stabilizer of some edge in the ith orbit. The quotient T/Γ is the graph used in defining Γ.

Recall from [1], VII.7–VII.9 that for any Γ-CW-complex X, there is a spectral sequence, with E_{1}^{p,q} = ⨁_{σ} H^{q}(Γ_{σ}), where the sum is over a set of orbit representatives of p-cells in X. For coefficients in a ring with trivial Γ-action (such as the field of p elements), this is a spectral sequence of rings. When X is acyclic the spectral sequence converges to a filtration of H^{p+q}(Γ). We apply this spectral sequence in the case when X = T. In this case

E_{1}^{0,q} ≅ H^{q}(P), E_{1}^{1,q} ≅ ⨁_{i=1}^{N} H^{q}(Q_{m(i)}),

and E_{1}^{p,q} = 0 for p > 1. Under this isomorphism the differential d_{1} : E_{1}^{0,q} → E_{1}^{1,q} has ith coordinate Res_{Q_{m(i)}}^{P} − ϕ_{i}^{∗}, and so E_{2}^{0,∗} is isomorphic to I(P, C). The fact that E_{2}^{p,q} = 0 for p > 1 implies that the spectral sequence collapses at the E_{2}-page. The edge homomorphism from E_{∞}^{0,∗} to H^{∗}(P) may be identified with Res_{P}^{Γ} (consider the map of spectral sequences induced by the inclusion of the vertex set of the tree in the whole tree, viewed as a map of Γ-spaces), and so a) is proved. For b), note that since E_{2}^{p,q} = 0 for p > 1, elements of E_{2}^{1,∗} uniquely determine elements of H^{∗}(Γ), and the product of any two such elements is zero in H^{∗}(Γ). Since Ker(Res_{P}^{Γ} ) may be identified with E_{2}^{1,∗}, b) follows, and c) follows immediately from b). Finally, d) follows from Theorem 5 stated above.

Remarks. 1) There are alternatives to using the equivariant cohomology spectral sequence in the proof of Theorem 1, but following a suggestion of the referee we decided to explain just one method in detail in the proof. Since the spectral sequence has only two non-zero rows it is essentially just a long exact sequence. This long exact sequence may be obtained by applying H^{∗}(Γ; ·) to the augmented chain complex for the tree T, modulo an application of the Eckmann-Shapiro lemma. We felt, however, that the ring structure of H^{∗}(Γ) is more easily understood in terms of the spectral sequence.

2) We believe that I(P, C_{u}) has some advantages over I(P, C_{s}). Both of these rings enjoy the finiteness property stated in Corollary 2. To compute I(P, C_{s}) one needs to know something about the p-local structure of all groups with Sylow subgroup P, whereas I(P, C_{u}) requires only knowledge of P.

3) On the other hand, I(P, C_{u}) does not retain much information concerning P. Let W (P) be the variety of all ring homomorphisms from I(P, C_{u}) to an algebraically closed field k of characteristic p. Then W (P) is determined up to homeomorphism by the p-rank of P: If P has p-rank n, then W (P) is homeomorphic to k^{n}/GL_{n}(F_{p}), and if E is an elementary abelian subgroup of P of rank n, then the induced map from W (E) to W (P) is an homeomorphism. These assertions concerning W (P) follow easily from Quillen’s theorem describing the variety of homomorphisms from H^{∗}(P) to k (see for example [5], chap. 9). Note that this the only place where we use Quillen’s theorem.

4) The definitions and theorems that we state remain valid if P is any finite group. We restrict to the case when P is a p-group only because this is the case occurring naturally in the work of Cartan-Eilenberg and Evens-Priddy.

5) The reader may have noticed that Theorems 1 and 3 work perfectly well for cohomology with coefficients in any ring R (viewed as a trivial P -module). Corollary 2 is valid for cohomology with coefficients in any ring R for which the Evens-Venkov theorem holds (see [5], 7.4 for a general statement).

6) The easiest way to relax the restrictions on the category C is to consider arbitrary finite categories with objects finite groups and morphisms injective group homomorphisms (it is unhelpful to view the groups as subgroups of a single group if the inclusion maps are not in the category). Define I(C) to be the limit over this category and for any group Γ, define D(Γ) to be the category of finite subgroups of Γ, with morphisms inclusions and conjugation by elements of Γ. Then one obtains

Theorem 1′. Let C be a connected finite category of finite groups and injective homomorphisms. Then there exists a discrete group Γ and a natural transformation from C to D(Γ) such that Γ and the induced map from H^{∗}(Γ) to I(C) satisfy properties a) to d) of Theorem 1.

Recall that a category is said to be connected if the equivalence relation on objects generated by ‘there is a morphism from Q to Q′’ has exactly one class. Note that there cannot be a direct analogue of Theorem 1 unless the category C is connected, since the degree zero part of I(C) is an F_{p} vector space of dimension the number of components of C, whereas H^{0}(Γ) ≅ F_{p}. The proof of Theorem 1′ is very similar to the proof of Theorem 1, except that one creates a graph of groups with one vertex for every object of C. The restriction to connected categories is not serious, since given any category C as above, one may make a connected category C^{+} by adding a trivial group to C as an initial object (i.e., add one new object, a trivial group, and one morphism from this object to every other object). The natural map from I(C^{+}) to I(C) is an isomorphism, except in degree zero.

The analogue of Corollary 2 in this generality, for which C need not be assumed to be connected, is:

Corollary 2′. Let C be a finite category of finite groups and injective homomorphisms. Then ∏_{Q∈C} H^{∗}(Q) is a finite module for I(C).

7) The following instance of Theorem 1 seems worthy of special note. Let P be an elementary abelian 2-group of rank n, let C be the category whose only object is P and whose morphisms are the group GL(n, F_{2}). Then H^{∗}(BΓ) is a ring whose radical is invariant under the action of the Steenrod algebra, and H^{∗}(BΓ)/\sqrt{0} is isomorphic to the Dickson algebra D_{n} = F_{2}[x_{1}, …, x_{n}]^{GL(n,F_{2})}. On the other hand it is known that for n ≥ 6, D_{n} itself cannot be the cohomology of any space [7].

Acknowledgements. This work was done when the authors met at the Centre de Recerca Matemàtica. The authors gratefully acknowledge the hospitality and support of the CRM. The first- and second-named authors were supported by E.C. Leibniz Fellowships, at the Max-Planck-Institut für Mathematik and at the CRM respectively.

References.

[1] K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87, Springer-Verlag, Heidelberg (1982).

[2] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton (1955).

[3] W. Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics 17, Cambridge Univ. Press, Cambridge (1989).

[4] L. Evens and S. Priddy, The ring of universally stable elements, Quart. J. Math. Oxford (2) 40 (1989), 399–407.

[5] L. Evens, Cohomology of groups, Oxford Mathematical Monographs, Clarendon Press, New York (1991).

[6] J.-P. Serre, Trees, Springer-Verlag, Heidelberg (1980).

[7] L. Smith and R. M. Switzer, Realizability and non-realizability of the Dickson algebras as cohomology rings, Proc. Amer. Math. Soc. 89 (1983), 303–313.

I. J. Leary, Max-Planck-Institut für Mathematik, 53225 Bonn, Germany.

From Jan. 1996: Faculty of Math. Studies, Univ. of Southampton, Southampton SO17 1BJ, England. B. Schuster, CRM, Institut d’Estudis Catalans, E-08193 Bellaterra (Barcelona), Spain.

From Jun. 1996: Fachbereich 7 Mathematik, Bergische Universität Wuppertal, Gaußstr. 20, 42097 Wuppertal, Germany.

N. Yagita, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan.