Proceedings of Symposia in Pure Mathematics Volume 00, 1997

A Variant of the Proof of the Landweber Stong Conjecture

Hans–Werner Henn

0. Let G be a finite group, k a finite field of order q, V an n - dimensional k[G]

- module and let S[V]^{G} be the graded ring of invariants with respect to the action of G on the symmetric algebra of V (where V is given grading 2 if q is odd and grading 1 otherwise). S[V]^{G} is a module over the Dickson algebra D(V) := S[V]^{Aut V} which is known to be a polynomial algebra on the Dickson generators c_{1}, ⋯, c_{n} of degree ϵ(q^{n} − q^{n−i}) with ϵ = 2 if q is odd and ϵ = 1 otherwise. The reader is referred to [W] for more information on the Dickson algebra. In [LSt] Landweber and Stong stated the following conjecture.

Landweber Stong Conjecture. The depth of S[V]^{G} is the maximal k such that c_{1}, ⋯, c_{k} is a regular sequence on S[V]^{G}.

We recall that a sequence x_{1}, ⋯, x_{n} of elements of positive degree in S[V]^{G} is called regular if multiplication by x_{1} is injective on S[V]^{G}, and for i > 1 multiplication by x_{i} is injective on S[V]^{G}/(x_{1}, ⋯, x_{i−1}); the depth of S[V]^{G} is defined to be the supremum of the lengths of all possible regular sequences.

The conjecture has recently been proved by Bourguiba and Zarati [BZ] in case k = 𝔽_{p} is a prime field. Their proof uses deep results from the theory of unstable modules over the mod - p Steenrod algebra A_{p}, most notably results on Lannes’ T - functor and the related functor Fix. The purpose of this note is to give a variant of their proof which we believe to be a bit simpler (but which also depends heavily on properties of Lannes’ T - functor). For background information on unstable modules and unstable algebras over the Steenrod algebra the reader is referred to [S]. Both proofs can be generalized to arbitrary finite fields and we will also make a few remarks on this at the end of the paper. Details of this generalization (in the case of the proof given in [BZ]) are given in [B].

I would like to thank Nick Kuhn, J. Lannes and S. Zarati for conversations on the material presented here

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

1991 Mathematics Subject Classification. Primary 13A50; Secondary 13C15, 55S10.

The author was supported by Deutsche Forschungsgemeinschaft and by Max-Planck-Institut für Mathematik at Bonn.

1. We fix a prime p and an unstable algebra K over A_{p} which is finitely generated as a graded 𝔽_{p} - algebra (we will say that K is an unstable noetherian algebra) and we are concerned with the category K_{fg} − U of unstable finitely generated K - modules. We recall that the objects of this category are unstable A_{p} - modules together with a map of unstable modules K ⊗ M ⟶ M which defines on M a graded K - module structure; furthermore M is assumed to have a finite number of K - module generators. The morphisms in K_{fg} − U are maps of graded 𝔽_{p} - vector spaces which commute both with the action of K and of A_{p}. For more on this category the reader may consult [H2].

Let d = d(K) denote the transcendence degree of K: by this we mean the Krull dimension of K as a graded 𝔽_{p} - algebra, or equivalently the maximum of the cardinalities of all sets of algebraically independent homogeneous elements. We assume that there are elements c_{i,K}, i = 1, …, d, in K with the following property: there exists a natural number e such that for every homomorphism φ of unstable algebras from K to the mod - p cohomology H^{∗}W of an elementary p - group W (i.e. W ≅ (ℤ/p)^{n} for some n, called the rank of W and also denoted by rk W) which makes H^{∗}W into a finitely generated K - module we have

φ(c_{i,K}) = \{ \begin{matrix}(c_{i,W})^{p^{e+d−rk W}} & if i ≤ rk W, \\ 0 & if i > rk W. \end{matrix}

Here c_{i,W} is the i - th Dickson invariant in H^{∗}W (of degree 2(p^{rk W} − p^{rk W −i}) if p is odd resp. of degree 2^{rk W} − 2^{rk W −i} if p = 2). We note that the finite generation assumption on H^{∗}W assures that rk W ≤ d.

In fact, for each unstable noetherian algebra K one can find such elements c_{i,K} and they are unique up to the process of taking iterated p - th powers. This can be deduced from the work of Rector [R] and Lam [Lm1, Lm2] and was apparently known to Landweber/Stong and Wilkerson in the 1980’s; it is also implicit in [HLS1]. For a detailed proof we refer to the appendix by J. Lannes in [BZ]. If K = H^{∗}W or K = D(W^{∗}) (with W^{∗} denoting the dual of W) then the elements c_{i,W} are well-known to qualify as elements c_{i,H^{∗}W} and also as elements c_{i,D(W^{∗})} (in both cases we have e = 0!).

After these preparations we can now state a theorem which in the case of prime fields generalizes the conjecture of Landweber and Stong. (Take K = S[V]^{Aut(V)} and M = S[V]^{G}!) As before we assume that all elements in regular sequences are homogeneous of positive degree and depth is defined in terms of such sequences.

Theorem 1. Let K be an unstable noetherian algebra of transcendence degree d and let M be an unstable finitely generated K- module. Then

depth M = max {0 ≤ r ≤ d| c_{1,K}, …, c_{rK} is a regular sequence on M} .

This result was proved recently by Bourguiba and Zarati [BZ]. Their proof as well as ours depends crucially on the progress in the theory of unstable A_{p} - modules within the last 10 years; in particular they depend heavily on the work of Lannes [Ln] and the earlier work of Lannes/Schwartz [LS] and Lannes/Zarati [LZ]. The first chapter of the recent book of Schwartz [S] provides more than adequate background on these prerequisites.

Beyond this we believe that the proof given here is more direct and represents a simplification over the original proof by Bourguiba and Zarati. It may therefore be of some interest.

2. We will follow Bourguiba and Zarati and call an unstable finitely generated K - module M for which the conclusion of the theorem holds an LS - module. The following result is the key-observation to the proof of [BZ] and also to our proof.

Proposition 2. Let K and M be as in the theorem. Then M can be embedded into an LS - module M′ with depth M′ ≥ depth M.

Proof of Theorem 1. (assuming Proposition 2) It is enough to show that if depth M ≥ r then c_{1,K}, …, c_{r,K} is a regular sequence on M and we will do this by induction on r. The case r = 0 is trivial.

So assume depth M = r > 0 and consider the short exact sequence

0 ⟶ M ⟶ M′ ⟶ M′′ ⟶ 0

where the embedding of M into M′ is as in Proposition 2. Because depth can be characterized in terms of non-vanishing of certain Ext - groups ([M, Thm. 28, p.100f]) we obtain depthM′′ ≥ r − 1, so c_{1,K}, ..., c_{r−1,K} is a regular sequence for M′′ by induction hypothesis, and hence M′′ is free as a module over the polynomial subalgebra 𝔽_{p}[c_{1,K}, ..., c_{r−1,K}] ⊂ K ([E, Prop. 10.3.4]). Therefore the exact sequence splits if it is considered as a sequence of modules over this polynomial algebra and consequently we still have an injection after dividing out the sequence c_{1,K}, ..., c_{r−1,K}, i.e. M/(c_{1,K}, ..., c_{r−1,K}) injects into M′/(c_{1,K}, ..., c_{r−1,K}). Because M′ is an LS - module with depth ≥ r the latter is torsion free with respect to c_{r,K} and therefore the former is as well and we are done. □

3. We now turn towards the proof of Proposition 2 and we start by describing certain objects of K_{fg} − U which are easily seen to be LS - modules.

So let φ : K ⟶ H^{∗}W be a homomorphism of unstable algebras which makes H^{∗}W into a finitely generated module over K and consider the corresponding “connected component” T_{W} (K; φ) of the unstable algebra T_{W} (K). We recall that T_{W} is a functor from the category U_{p} of unstable modules to itself which is left adjoint to tensoring with H^{∗}W. In fact, T_{W} even lifts to a functor from unstable algebras to itself and continues to have the adjointness property in this context. In particular the adjoint of φ is a map T_{W} K ⟶ 𝔽_{p} of unstable algebras. Then T_{W} (K; φ) is defined as the “component” T_{W} K ⊗_{T_{W}^{0} K} 𝔽_{p}(φ) where T_{W}^{0} denotes the homogeneous part of degree 0 of T_{W} and 𝔽_{p}(φ) denotes the T_{W}^{0} K - module 𝔽_{p} with module structure given by the adjoint of φ.

We will need to consider the map φ̃: K ⟶ H^{∗}W ⊗ T_{W} (K; φ) which is adjoint to the projection T_{W} K ⟶ T_{W} (K; φ). This map has the form φ̃k = φk ⊗ 1 + y where y is a sume of terms which have positive degree in the second tensor variable. Now let F be an unstable T_{W} (K; φ) - module which is finite as a graded vector space and consider H^{∗}W ⊗ F as an unstable K - module via the K - module structure which is pulled back from the obvious H^{∗}W ⊗ T_{W} (K; φ) - module structure via φ̃. The proof of the following result is straightforward, say by filtering H^{∗}W ⊗ F by the degree in the second tensor factor so that the associated graded object has its K - module structure given via φ only; we leave this to the reader.

Lemma 3. The unstable K - module H^{∗}W ⊗F described above has the following properties.

a) H^{∗}W ⊗ F is finitely generated as K - module.

b) The sequence c_{1,K}, … c_{rkW,K} is a regular sequence on H^{∗}W ⊗ F. Furthermore depth(H^{∗}W ⊗ F) = rk W; in particular H^{∗}W ⊗ F is an LS - module.

The following result is now the crucial input from the theory of unstable modules. A geodesic to its proof for somebody familiar with chapter I of [S] should be given by section 1 of [H1] and section 1 of [H2].

Theorem 4 ([HLS2, Thm I.4.9], [H2, Thm 1.9]). For any unstable finitely generated K - module M there exists an embedding into a finite direct sum ⨁_{α} H^{∗}W_{α} ⊗ F_{α} where each summand is a module as in Lemma 3.

Proof of Proposition 2. Assume that depthM = r. We will take an embedding M ↪ ⨁_{α} H^{∗}W_{α} ⊗ F_{α} as provided by Theorem 4 and we will assume that this embedding is reduced in the sense that no summand can be deleted without losing the embedding property. It suffices to show that in such a reduced embedding all summands will have rk W_{α} ≥ r.

Otherwise we would find x ∈ M and some α with rk W_{α} < r such that the unstable K - submodule M_{x} generated by x embedds into H^{∗}W_{α}⊗F_{α}. In particular, the order of the pole at t = 1 of the Poincaré series χ(M_{x}) := Σ_{n} dim_{𝔽_{p}}(M_{x})^{n}t^{n} is equal to rkW_{α} < r. On the other hand M has depth r, hence ([E, Prop. 10.3.4]) it is a free module over a polynomial algebra 𝔽_{p}[x_{1}, … x_{r}] whose generators x_{i} form a regular sequence on M and are of positive degree. M_{x} is a non - trivial submodule in M and hence χ(M_{x}) dominates from above χ(J) where J is a non-trivial principal ideal inside this polynomial algebra, namely the ideal generated by a non-trivial component of x if x is expressed as a linear combination of the basis elements of the free module M. Now a principal ideal over an integral domain is always free on its generator, hence the order of the pole at t = 1 of χ(J) is equal to r, i.e. the order of the pole at t = 1 of χ(M_{x}) is at least r and we arrive at a contradiction.□

4. The proof of the Landweber-Stong-Conjecture in the case of an arbitrary field follows the same strategy as in the prime field case. In particular this involves finding appropriate definitions of the Steenrod algebra, the categories K − U, the T - functor etc. in the case of an arbitrary finite field. Most of these concepts have been already developed by N. Kuhn in [K1, K2]. Once the basic definitions have been set up one can verify that the proof given here resp. in [BZ] can be carried over. More details on this can be found in [B].

References

[B] D. Bourguiba, Profondeur et algèbre de Steenrod, Thèse, Université de Tunis 1997.

[BZ] D. Bourguiba and S. Zarati, On the depth of an unstable H^{∗}V - A - module which is of finite type as an H^{∗}V - module, to appear in Invent. Math..

[E] L. Evens, The cohomology of groups, Oxford Mathematical Monographs, Clarendon Press, [H1] H.-W. Henn, Finiteness properties of injective resolutions of certain unstable modules

over the Steenrod algebra and applications, Math. Ann. 291 (1991), 191-203.

[H2] H.-W. Henn, Commutative algebra of unstable K - modules, Lannes’ T - functor and equivariant mod - p - cohomology, Journal für die reine und angewandte Mathematik 478 (1996), 189-215.

[HLS1] H.-W. Henn, J. Lannes and L. Schwartz, The categories of unstable modules and unstable algebras modulo nilpotent objects, American Journal of Math. 115 (1993), 1053-1106.

[HLS2] H.-W. Henn, J. Lannes and L. Schwartz, Localization of unstable A - modules and equivariant mod - p cohomology, Math. Ann. 301 (1995), 23-68.

[K1] N. Kuhn, Generic representation theory of the general linear groups and the Steenrod algebra: I, American Journal of Math. 116 (1994), 327–360.

[K2] N. Kuhn, Generic representation theory of the general linear groups and the Steenrod algebra: III, K-Theory 9 (1995), 273–303.

[Lm1] S. P. Lam, Unstable algebras over the Steenrod algebra and cohomology of classifying spaces, Thesis, Univ. of Cambridge (1982).

[Lm2] S. P. Lam, Unstable algebras over the Steenrod algebra, Springer Lect. Notes in Math. 1051, 1984, pp. 374–392.

[LSt] P. S. Landweber and R. E. Stong, The depth of rings of invariants over finite fields, Springer Lect. Notes in Math. 1240, 1987, pp. 259-274.

[Ln] J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire, Publ. Math. IHES 75 (1992), 135-244.

[LS] J. Lannes and L. Schwartz, Sur la structure des A - modules instables injectifs, Topology 28 (1989), 153-169.

[LZ] J. Lannes and S. Zarati, Sur les U-injectifs, Ann. Scient. Ec. Norm. Sup. 19 (1986), 303-333.

[M] H. Matsumura, Commutative Algebra, 2nd ed. Math. Lect. Notes series. Benjamin, 1980.

[R] D. Rector, Noetherian cohomology rings and finite loop spaces with torsion, J. P. A. A. 32 (1984), 191-217.

[S] L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, University of Chicago Press, 1994.

[W] C. Wilkerson, A primer on the Dickson invariants, Contemporary Mathematics, Proc. of the Northwestern Homotopy Theory Conference (1983), 421-434.

Max-Planck-Institut für Mathematik, Gottfried Claren Straße 26, D–53225 Bonn, Fed. Rep. of Germany

    E-mail address: henn@mpim-bonn.mpg.de