NILPOTENCE IN GROUP COHOMOLOGY

NICHOLAS J. KUHN

Abstract. We study bounds on nilpotence in H^{∗}(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds recent verification of Dave Benson’s Regularity Conjecture. New results are given for finite p–groups, leading to good bounds on nilpotence in H^{∗}(BP) determined by the subgroup structure of the p–group P.

1. Introduction

Fixing a prime p, let H^{∗}(BG) denote the mod p cohomology ring of the classifying space of a compact Lie group G. This is a graded commutative 𝔽_{p}–algebra of great interest as it is the home for mod p characteristic classes of principal G bundles. Furthermore, when G is finite, this ring identifies with Ext_{𝔽_{p}[G]}^{∗}(𝔽_{p}, 𝔽_{p}), and so contains much detailed module theoretic information.

Precise calculation of H^{∗}(BG) can be daunting, particularly when G is a finite p–group. In this paper we study nilpotence in H^{∗}(BG). We offer some updates of our previous work in [K3], together with new results in the finite p–group case.

We should be more precise about what we mean by ‘nilpotence’.

Let Rad(G) be the nilradical of the graded 𝔽_{p}–algebra H^{∗}(BG). One can define an ‘algebraic’ nilpotence degree as follows.

Definition 1.1. Define d^{alg}(G) to be the maximal d such that Rad(G)^{d} ≠ 0.

As the mod p cohomology of a topological space, H^{∗}(BG) is in the category U, the category of modules over the mod p Steenrod algebra A_{p} which satisfy the unstable condition. Following Hans-Werner Henn, Jean Lannes, and Lionel Schwartz in [HLS], one can define a ‘topological’ nilpotence degree as follows. Let Σ^{d}M denote the dth suspension (upward shift) of a graded module M.

Definition 1.2. Define d^{U}(G) to be the maximal d such that H^{∗}(BG) contains a nonzero submodule of the form Σ^{d}M, with M ∈ U.

This definition is clearly just dependent on the A_{p} module structure of H^{∗}(BG), but a result in [HLS] allows for comparison with d^{alg}(G). As will be reviewed in §2, d^{U}(G) is also the minimal d such that the algebra homomorphism

q_{d} : H^{∗}(BG) → ∏_{V} H^{∗}(BV) ⊗ H^{≤d}(BC_{G}(V))

is monic. Here the product is over the elementary abelian p–subgroups of G, the component maps are induced by the the group homomorphisms V × C_{G}(V) → G, and M^{≤d} denotes the quotient of a graded module M by all elements of degree more than d.

Date: February 21, 2010.

2000 Mathematics Subject Classification. Primary 20J06; Secondary 55R40.

This research was partially supported by a grant from the National Science Foundation.

One easily concludes (see Corollary 2.6) that

d^{alg}(G) ≤ \begin{cases}d^{U}(G) & if p = 2 \\ d^{U}(G) + r(G) & if p is odd. \end{cases}

Here r(G) is the maximal rank of an elementary abelian p–subgroup of G.

Our goal here is to describe how to calculate d^{U}(G), and, in particular, to give good group theoretic upper bounds. We note that d^{U}(ℤ/p) = 0 and d^{U}(G × H) = d^{U}(G) + d^{U}(H). Furthermore, by transfer arguments, d^{U}(G) ≤ d^{U}(P), if P is a p–Sylow subgroup of a finite group G, and a similar inequality holds for a general compact Lie group G, with P now the evident extension of a maximal torus T by a p–Sylow subgroup of N_{G}(T)/T.

1.1. A general bound on d^{U}(G). We recall some terminology from [K3].

Notation 1.3. Let C(G) < G be the maximal central elementary abelian p– subgroup, and let c(G) denote its rank. Via restriction, H^{∗}(BC(G)) is a finitely generated H^{∗}(BG)–module, and we let e(G) denote the top degree of a generator. Finally, here and throughout the paper, we let V denote an elementary abelian p–group, i.e. a group isomorphic to (ℤ/p)^{r} for some r.

Theorem 1.4. If G is compact Lie, then

max_{\begin{matrix}V <G \\ r(V)=r(G) \end{matrix}}\{e(C_{G}(V)) − dim(C_{G}(V))\} ≤ d^{U}(G) ≤ max_{V <G}\{e(C_{G}(V)) − dim(C_{G}(V))\}.

Here dim(G) denotes the dimension of a Lie group G as a manifold, and so is 0 if G is finite. Note that the lower bound equals the upper bound when G is p–central – a group in which every element of order p is central – and, in that case, d_{0}^{U}(G) = e(G) − dim G.

The proof of this theorem is given in §2. Most of this is a review and slight reorganization of work in [K3], with results extended to all compact Lie groups. Some of our results were previously conditional on the verification of Dave Benson’s Regularity Conjecture [B2] which conjectured the vanishing of certain local cohomology groups. Happily, this is now a theorem of Peter Symonds [Sy], and we make very precise how the vanishing of local cohomology groups allows for improvement on Theorem 1.4.

1.2. Bounds for finite p–groups. Further investigations when G is a finite p– group lead to some very nice general statements.

The first is that the upper bound given in Theorem 1.4 simplifies.

Theorem 1.5. If P is a p–group, then d^{U}(P) ≤ e(P).

We then have two new estimates of e(P).

Theorem 1.6. Suppose a p–group P acts faithfully on a set S with no fixed points. Then

e(P) ≤ \begin{cases}|S|/2 − |S/P | & if p = 2 \\ 2|S|/p − |S/P | & if p is odd. \end{cases}

Here |S| is the cardinality of S.

Theorem 1.7. Let A < P be an abelian subgroup of maximal order in a p–group P. Then e(P) ≤ c(P)(2|P |/|A| − 1).

Example 1.8. Both of these last two theorems are nicely illustrated by the following example. Let P be a 2–Sylow subgroup of the finite group SU(3, 4). P is a 2–central group of order 64, of exponent 4, with C(P) = [P, P] ≃ ℤ/2 × ℤ/2: see [K2, §6.3] for a useful description of this group. Both theorems give us the estimate e(P) ≤ 14, which, in fact, computation shows equals e(P), and thus d^{U}(P).

To use Theorem 1.6, let a, b ∈ P be elements of order 4 with a^{2} ≠ b^{2}. Then P acts faithfully on S = P/⟨a⟩ ∐ P/⟨b⟩ with no fixed points, so e(P) ≤ 32/2−2 = 14.

To use Theorem 1.7, the centralizer of any element of order 4 is isomorphic to ℤ/4 × ℤ/4, thus e(P) ≤ 2[2(64/16) − 1] = 14.

The proofs of both Theorem 1.5 and Theorem 1.6 depend on the following monotonicity result, which at first surprised us, as it is false for arbitrary finite groups.

Theorem 1.9. Let Q be a subgroup of a p–group P. Then e(Q) ≤ e(P).

Theorem 1.7 is proved using Chern classes of representations, and would be a special case of the next conjecture, where we let n(G) denote the minimal dimension (over ℂ) of a faithful complex representation of G.

Conjecture 1.10. If G is compact Lie, then e(G) ≤ 2n(G) − c(G).

Assuming this, one easily deduces the general estimate d^{U}(G) ≤ 2n(G) − c(G): see Remark 3.9. This should be compared to the estimate in [HLS]: d^{U}(G) ≤ n(G)^{2}.

§3 contains the proofs of all of these theorems and a discussion of the conjecture. The most subtle argument is the proof of Theorem 1.9. Proved by induction on the order of P, it reduced to a problem about invariants of arbitrary ℤ/p actions on subHopf algebras of polynomial algebras over 𝔽_{p}: see Problem 3.11. This we then deal with in §4, proving results which may be of independent interest.

Remark 1.11. We note that our paper [K3] has tables of examples made using the Jon Carlson’s cohomology website [Ca2]. Thousands more examples are now similarly accessible using the cohomology website of David Green and Simon King [GK]. Their implementation includes the calculation of the restriction of H^{∗}(BP) to H^{∗}(BC(P)), so that e(P) can be immediately read off of their data. For example, one see that if P is the 2–Sylow subgroup of the third Conway group, so P has order 1024, then e(P) = 7 and so, combining Theorem 1.5 with the fine points of Theorem 2.22, we see that d^{U}(P) ≤ 6.

1.3. Acknowledgements. The organization of §2 follows the presentation I gave at the Conference on Algebraic Topology, Group Theory, and Representation Theory held on the Isle of Skye, Scotland in June, 2009. I have tried to keep the audience I had there in mind. Conversations with Benson, Symonds, and invariant theorists David Wehlau, Jim Shank, and Eddy Campbell have been helpful.

2. Old results revisited

In this section, we prove the bounds for d^{U}(G) given in Theorem 1.4. The main steps are as follows, where terminology and notation will be defined in due course:

• d^{U}(G) = max {d_{0}(Cess^{∗}(BC_{G}(V))) | V < G}. See Proposition 2.8.

• d_{0}(Cess^{∗}(BG)) = e_{prim}(G). See Corollary 2.14.
• e_{prim}(G) ≤ e_{indec}(G). See Corollary 2.20.
• e_{indec}(G) ≤ e(G) − dim G. See Theorem 2.22 for this and a bit more.

This last inequality refines using local cohomology as follows:

• e_{indec}(G) = e(G) + max{e | H_{𝔪}^{c(G),−c(G)+e}(H^{∗}(BG)) ≠ 0}.

See Theorem 2.29.

• H_{𝔪}^{s,t}(H^{∗}(BG)) = 0 if s + t > − dim G. This is Symonds’ theorem [Sy].

2.1. The basic ring structure of H^{∗}(BG). We begin by recalling a fundamental example. If V = (ℤ/p)^{r}, then

H^{∗}(BV) ≃ \begin{cases}𝔽_{2}[x_{1}, …, x_{r}] & if p = 2 \\ Λ(x_{1}, …, x_{r}) ⊗ 𝔽_{p}[y_{1}, …, y_{r}] & if p is odd, \end{cases}

where |x_{i}| = 1 and y_{i} = β(x_{i}). (β is the Bockstein homomorphism.) Furthermore, addition V × V → V induces a primitively generated Hopf algebra structure on H^{∗}(BV).

More generally, H^{∗}(BG) can be difficult to compute explicitly, particularly when G is a more interesting finite p–group. For example, if P is the 2–Sylow subgroup of SU_{3}(4), as in Example 1.8, a minimal presentation of the algebra H^{∗}(BP) has 26 generators (in degrees up to 11) and 270 relations (in degrees up to 22). (See [CTVZ, group #187], or [GK, group #145].)

In spite of this, some basic ring structure has been known for a long time. In the late 1960’s [Q] D.Quillen showed that H^{∗}(BG) is Noetherian of Krull dimension r(G); equivalently, H^{∗}(BG) is a finitely generated module over a subpolynomial algebra on r(G) generators. A decade later J.Duflot [D] showed that its depth is at least c(G); equivalently, H^{∗}(BG) is a free module over a subpolynomial algebra on c(G) generators.

Remark 2.1. The extreme situation, when c(G) = r(G), happens precisely when G is p–central. Then H^{∗}(BG) will be Cohen–MacCauley: the depth of H^{∗}(BG) will equal its Krull dimension. In general, there is no group theoretic criterion characterizing either groups G such that the depth of H^{∗}(BG) equals the lower bound c(G), or groups G such that the depth of H^{∗}(BG) equals the upper bound r(G).

Quillen’s idea was to probe H^{∗}(BG) by its restrictions to its elementary abelian p–subgroups. The product over all such restrictions gives a ring homomorphism

q_{0} : H^{∗}(BG) → ∏_{V <G} H^{∗}(BV).

Recall that, given K < G, the restriction map H^{∗}(BG) → H^{∗}(BK) makes H^{∗}(BK) into a finitely generated H^{∗}(BG)–module. Thus the codomain of q_{0}, a ring whose Krull dimension is clearly r(G), is finitely generated over H^{∗}(BG). Quillen then shows that ker(q_{0}) is nilpotent, which then immediately implies the result about Krull dimension.

2.2. The nilpotent filtration of U. As the mod p cohomology of a topological space, H^{∗}(BG) is an unstable algebra over the mod p Steenrod algebra A_{p}. We recall that an A_{p}–module M is unstable, if, when p = 2, Sq^{k}x = 0 if k > |x|, and, when p is odd, β^{e}P^{k}x = 0 if 2k+e > |x|. M is an unstable algebra if in addition, it is a graded commutative algebra satisfying both the Cartan and Restriction formulae.

The 1980’s featured much remarkable work on K and U, the categories of unstable algebras and modules, with the algebras H^{∗}(BV) playing a special role. (See [S2] for entry into the extensive literature.)

In the 1995 paper [HLS], H.-W.Henn, J.Lannes, and L.Schwartz revisited Quillen’s work from this new perspective. Following [HLS], we have the following definition.

Definition 2.2. If M is an unstable A_{p}–module, let d_{0}(M) be the maximal d such that M contains a nonzero submodule of the form Σ^{d}N, with N unstable. If no such maximum exists, let d_{0}(M) = ∞, and let d_{0}(0) = −∞.

Thus the invariant d^{U}(G) of the introduction is d_{0}(H^{∗}(BG)).

An alternate definition, easily shown equivalent to the one above, is that d_{0}(M) is the length of the nilpotent filtration [S1] of M,

⋯ ⊂ nil_{d}M ⊂ nil_{d−1} ⊂ nil_{1}M ⊂ nil_{0}M = M,

where nil_{d}M is the large submodule in the localizing subcategory of U generated by the d–fold suspensions.

Three elementary properties of d_{0}(M) are stated in the next lemma.

Lemma 2.3. (a) If M is nonzero in degree d, but zero in all higher degrees, then d_{0}(M) = d.

(b) If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence in U, then d_{0}(M_{1}) ≤ d_{0}(M_{2}), and d_{0}(M_{2}) ≤ max{d_{0}(M_{1}), d_{0}(M_{3})}.

(c) d_{0}(H^{∗}(Bℤ/p)) = 0.

The next properties are considerably deeper. References for (a) are [K1, Prop.2.5] or [HLS, Prop.I.3.6] . Property (b) concerns Lannes’ functor [L2] T_{V} : U → U, the left adjoint to the functor M ⇝ H^{∗}(BV) ⊗ M, and a reference is [K3, Prop.3.12]. Property (c) is due to Henn [H].

Proposition 2.4. (a) d_{0}(M ⊗ N) = d_{0}(M) + d_{0}(N).

(b) d_{0}(T_{V} M) = d_{0}(M).

(c) d_{0}(M) < ∞ if M is a finitely generated module over an Noetherian unstable algebra K with structure map K ⊗ M → M in U.

2.3. The comparison between d^{alg}(G) and d^{U}(G). Note that property (c) of the last proposition implies that d^{U}(G) < ∞, so that the nilpotent filtration of H^{∗}(BG) has finite length.

In [HLS], the authors show how to generalize Quillen’s map q_{0} to realize the nilpotent filtration of H^{∗}(BG). For each d ≥ 0, let

q_{d} : H^{∗}(BG) → ∏_{V} H^{∗}(BV) ⊗ H^{≤d}(BC_{G}(V))

be the map of unstable algebras as defined in the introduction. They observe that ker q_{d} = nil_{d+1}H^{∗}(BG), and so we have the following.

Proposition 2.5. d^{U}(G) is the minimal d such that q_{d} is monic.

If I is a nilpotent ideal in a graded Noetherian ring, let d^{alg}(I) be the maximal d such that I^{d} ≠ 0. Thus the invariant d^{alg}(G) of the introduction is d^{alg}(Rad(G)). Note that

d^{alg}(H^{∗}(BV) ⊗ H̃^{≤d}(BC_{G}(V))) = \begin{cases}d^{alg}(H̃^{≤d}(BC_{G}(V)) & if p = 2 \\ d^{alg}(Λ(V^{#}) ⊗ H̃^{≤d}(BC_{G}(V))) & if p is odd. \end{cases}

Corollary 2.6. With d = d^{U}(G),

d^{alg}(G) ≤ \begin{cases}max_{V}\{d^{alg}(H̃^{≤d}(BC_{G}(V))\} ≤ d & if p = 2 \\ max_{V}\{d^{alg}(H̃^{≤d}(BC_{G}(V)) + r(V)\} ≤ d + r(G) & if p is odd. \end{cases}

2.4. Central essential cohomology. The following definition from [K3] is a variant of Carlson’s Depth Essential Cohomology [CTVZ].

Definition 2.7. Let Cess^{∗}(BG) be the kernel of the map

H^{∗}(BG) → ∏_{C(G)≨V} H^{∗}(BC_{G}(V)).

This is an unstable A–module. Cess^{∗}(BG) = H^{∗}(BG) exactly when the product is over the empty set, i.e. G is p–central. Cess^{∗}(BG) can also be zero: as we will see, Cess^{∗}(BG) ≠ 0 if and only if the depth of H^{∗}(BG) = c(G).

Proposition 2.8. d^{U}(G) = max {d_{0}(Cess^{∗}(BC_{G}(V))) | V < G}.

To prove this, we first need the following consequence of the calculation of T_{V} H^{∗}(BG) due to Lannes [L1].

Proposition 2.9. For all V < G, H^{∗}(BC_{G}(V)) is a summand of T_{V} H^{∗}(BG), and thus d^{U}(C_{G}(V)) ≤ d^{U}(G).

Proof of Proposition 2.8. This follows by downward induction on the rank of C(G). From the exact sequence

0 → Cess^{∗}(BG) → H^{∗}(BG) → ∏_{C(G)≨V} H^{∗}(BC_{G}(V)),

one sees that

d^{U}(G) ≤ max {d_{0}(Cess^{∗}(BG)), d^{U}(C_{G}(V)) | C(G) ≨ V < G}.

But this inequality is an equality by the last proposition. □

2.5. Primitives in central essential cohomology. For the rest of this section, we fix a compact Lie group G, and let C = C(G).

By an unstable H^{∗}(BC)–comodule, we will mean an unstable module M having an H^{∗}(BC)–comodule structure map Δ : M → H^{∗}(BC)⊗M that is in U. Examples of interest to us include H^{∗}(BG), H^{∗}(BC_{G}(V)) for all V < G, and Cess^{∗}(BG), where the comodule structures are all induced by the group homomorphism C×G → G.

Definitions 2.10. If M is an unstable H^{∗}(BC)–comodule, we define its associated module of primitives to be

P_{C}M = \{x ∈ M | Δ(x) = 1 ⊗ x\} = Eq \{M \begin{matrix}Δ \\ ⟶ \\ ⟶ \\ i \end{matrix} H^{∗}(C) ⊗ M\}.

If P_{C}M is finite dimensional, we let e_{prim}(M) be its largest nonzero degree, or −∞ if M = 0.

Note that P_{C}M is again an unstable module.

Lemma 2.11. If M is an unstable H^{∗}(BC)–comodule, and P_{C}M is finite dimensional, then d_{0}(M) = e_{prim}(M).

Proof. Assume P_{C}M is finite dimensional with largest nonzero degree e = e_{prim}(M). Then e = d_{0}(P_{C}M). Since P_{C}M is an unstable submodule of M, d_{0}(P_{C}M) ≤ d_{0}(M). Finally, the composite

M ⟶^{Δ} H^{∗}(BC) ⊗ M ↠ H^{∗}(BC) ⊗ M^{≤e}

will be monic, so that

d_{0}(M) ≤ d_{0}(H^{∗}(BC) ⊗ M^{≤e}) = d_{0}(M^{≤e}) = e.
□

Proposition 2.12. P_{C}Cess^{∗}(BG) is finite dimensional.

Proof. [K3, Thm.8.5] implies that if P_{C}Cess^{d}(BG) ≠ 0, then d ≤ d^{U}(G). □

Remark 2.13. The careful reader will discover that [K3, Thm.8.5] has a rather delicate proof, using related results in [K2], all based on careful analysis of formulae in [HLS]. It would be nice to have a simpler proof of the proposition. In the next subsection we will see (Corollary 2.19) that P_{C}Cess^{∗}(BG) is finite dimensional if and only if Cess^{∗}(BG) has Krull dimension equal to c(G). When G is finite, this Krull dimension calculation is verified [K3, Prop.8.2] using a result of J.Carlson [Ca1].

We let e_{prim}(G) denote e_{prim}(Cess^{∗}(BG)).

Corollary 2.14. d_{0}(Cess^{∗}(BG)) = e_{prim}(G).

2.6. Duflot algebras. Let c = c(G), the rank of C = C(G), so that

H^{∗}(BC) ≃ \begin{cases}𝔽_{2}[x_{1}, …, x_{c}] & if p = 2 \\ Λ(x_{1}, …, x_{c}) ⊗ 𝔽_{p}[y_{1}, …, y_{c}] & if p is odd. \end{cases}

The image of the restriction homomorphism i^{∗} : H^{∗}(BG) → H^{∗}(BC) will be a sub Hopf algebra of H^{∗}(BC). Thus, after a change of basis for H^{1}(BC), it will have the form

im(i^{∗}) = \begin{cases}𝔽_{2}[x_{1}^{2^{j_{1}}} , …, x_{c}^{2^{jc}}] & if p = 2 \\ 𝔽_{p}[y_{1}^{p^{j_{1}}} , …, y_{b}^{p^{j_{b}}} , y_{b+1}, …, y_{c}] ⊗ Λ(x_{b+1}, …, x_{c}) & if p is odd, \end{cases}

with the j_{i} forming a sequence of nonincreasing nonnegative integers. (See [BrH, Rem.1.3] and [AS].) In the odd prime case, c − b has group theoretic meaning as the rank of the largest subgroup of C splitting off G as a direct summand.

As in [K3], we will say that G has type [a_{1}, …, a_{c}] where

(a_{1}, …, a_{c}) = \begin{cases}(2^{j_{1}}, …, 2^{j_{c}}) & if p = 2 \\ (2p^{j_{1}}, …, 2p^{j_{b}}, 1, …, 1) & if p is odd. \end{cases}

Recall that e(G) is defined to be the largest degree of a H^{∗}(BG)–module generator of H^{∗}(BC), i.e. the top degree of the finite dimensional Hopf algebra H^{∗}(BC) ⊗_{H^{∗}(BG)} 𝔽_{p}. Note that this number is determined by the type of G:

e(G) = ∑_{i=1}^{c}(a_{i} − 1).

Since im(i^{∗}) is a free commutative algebra, one can split the epimorphism of rings i^{∗} : H^{∗}(BG) ↠ im(i^{∗}), and make the next definition.

Definition 2.15. A Duflot algebra of H^{∗}(BG) is a subalgebra A ⊆ H^{∗}(BG), such that i^{∗} : A → im(i^{∗}) is an isomorphism.

Remark 2.16. It seems unclear that a Duflot algebra can always be chosen to also be closed under Steenrod operations. Nor does it seem that it can be always chosen to be a sub–H^{∗}(BC)–comodule of H^{∗}(BG).

2.7. Indecomposables in central essential cohomology. For the rest of the section, now also fix a Duflot algebra A ⊆ H^{∗}(BG).

Definitions 2.17. If M is an A–module, we define the A–indecomposables to be Q_{A}M = M ⊗_{A} 𝔽_{p}. If Q_{A}M is finite dimensional, we let e_{indec}(M) be its largest nonzero degree, or −∞ if M = 0.

Observe that everything in the exact sequence

0 → Cess^{∗}(BG) → H^{∗}(BG) → ∏_{C≨V} H^{∗}(BC_{G}(V))

is both an A–module and a H^{∗}(BC)–comodule. These structures are sufficiently compatible ‘up to filtration’ that one can prove the following.

Proposition 2.18. The following hold.

(a) Cess^{∗}(BG) is a free A–module.

(b) The composite P_{C}Cess^{∗}(G) ↪ Cess^{∗}(G) ↠ Q_{A}Cess^{∗}(G) is monic.

(c) The sequence 0 → Q_{A}Cess^{∗}(G) → Q_{A}H^{∗}(G) → ∏_{C(G)<U} Q_{A}H^{∗}(C_{G}(U)) is

exact.

See [K3, Prop.8.1].

Corollary 2.19. Q_{A}Cess^{∗}(BG) is finite dimensional if and only if P_{C}Cess^{∗}(BG) is finite dimensional. In this case, e_{prim}(Cess^{∗}(BG)) ≤ e_{indec}(Cess^{∗}(BG)).

Proof. For notational simplicity, let M = Cess^{∗}(BG). The proposition immediately implies that if Q_{A}M is finite dimensional so is P_{C}M, and the stated inequality will hold. Conversely, suppose P_{C}M is finite dimensional. Recall that the composite (of A–modules)

M ⟶^{Δ} H^{∗}(BC) ⊗ M → H^{∗}(BC) ⊗ M^{≤e_{prim}(M)}

is monic. As H^{∗}(BC) ⊗ M^{≤e_{prim}(M)} is certainly a finitely generated A–module, so is M. □

We let e_{indec}(G) denote e_{indec}(Cess^{∗}(BG)).

Corollary 2.20. Cess^{∗}(BG) is a finitely generated free A–module, and e_{prim}(G) ≤ e_{indec}(G).

Remark 2.21. As we observed computationally in [K3, Appendix A], e_{prim}(G) = e_{indec}(G) for all finite 2–groups G of order dividing 32. We suspect that this pattern will not continue, but it would be nice to have an explicit example for which the inequality of the corollary is strict.

2.8. Local cohomology and Symond’s theorem. The last step in our proof of Theorem 1.4 is the verification of the next bound.

Theorem 2.22. For all G, e_{indec}(G) ≤ e(G) − dim G. The inequality is strict unless G is p–central. If G is p–central, then d_{U}(G) = e_{prim}(G) = e_{indec}(G) = e(G) − dim G.

We note that, even when p is odd, it suffices to prove this when the Duflot algebra A is a polynomial algebra, i.e. when G has no ℤ/p direct summands, as d_{U}(G × V) = d_{U}(G), e_{prim}(G × V) = e_{prim}(G), e_{indec}(G × V) = e_{indec}(G), and e(G × V) = e(G).

We need to begin with a quick summary of definitions and properties of local cohomology. A general reference for this is [BS].

Let 𝔪 be a maximal ideal in a graded Noetherian ring R. For M an R–module,

M ↦ H_{𝔪}^{s,∗}(M)

is defined to be the sth right derived functor of

M ↦ H_{𝔪}^{0,∗}(M) = the 𝔪–torsion part of M.

Proposition 2.23. H_{𝔪}^{s,∗}(M) ≠ 0 only if depth_{𝔪}M ≤ s ≤ dim M. Furthermore, If s = depth_{𝔪}M or s = dim M, then H_{𝔪}^{s,∗}(M) ≠ 0.

This is the content of [BS, Cor.6.2.8].

We need some related results about how local cohomology interacts with regular M–sequences. Let |z| denote the degree of z ∈ R.

Lemma 2.24. Fix (s, t), and suppose that H_{𝔪}^{s′,t′}(M) = 0 for s′ < s and for (s, t′) with t′ > t. If z ∈ R is an M–regular element, then H_{𝔪}^{s′,t′}(M/(z)) = 0 for s′ < s−1 and for (s − 1, t′) with t′ > t + |z|, and, furthermore

H_{𝔪}^{s−1,t+|z|}(M/(z)) ≃ H_{𝔪}^{s,t}(M).

Proof. By assumption, z is not a zero divisor of M, so there is a short exact sequence of R–modules

0 → Σ^{|z|}M ⟶^{z} M → M/(z) → 0.

The lemma then follows from the associated long exact sequence, which has the form

… → H_{𝔪}^{s′−1,t′+|z|}(M) → H_{𝔪}^{s′−1,t′+|z|}(M/(z)) → H_{𝔪}^{s′,t′}(M) → H_{𝔪}^{s′,t′+|z|}(M) → …

□

By induction on the length of a regular sequence, the lemma has the following corollary.

Corollary 2.25. With assumptions on (s, t) and M as in the lemma, if z_{1}, …, z_{s} is an M–regular sequence, then H_{𝔪}^{0,t′}(M/(z_{1}, …, z_{s})) = 0 for t′ > t+|z_{1}|+⋯+|z_{s}|, and

H_{𝔪}^{0,t+|z_{1}|+···+|z_{s}|}(M/(z_{1}, …, z_{s})) ≃ H_{𝔪}^{s,t}(M).

We now apply this in the case when R = M = H^{∗}(BG) and 𝔪 = H̃^{∗}(BG). Let c = c(G), r = r(G). If z_{1}, …, z_{c} are algebra generators for the Duflot algebra A, then |z_{1}| + ⋯ + |z_{s}| = c + e(G), and M/(z_{1}, …, z_{c}) = Q_{A}M, and the corollary tells us the following.

Proposition 2.26. Suppose H_{𝔪}^{c(G),−c(G)+e′}(H^{∗}(BG)) = 0 for all e′ > e. Then H_{𝔪}^{0,e(G)+e′}(Q_{A}H^{∗}(BG)) = 0 for all e′ > e, and

H_{𝔪}^{0,e(G)+e}(Q_{A}H^{∗}(BG)) = H_{𝔪}^{c,−c+e}(H^{∗}(BG)).

Now we note

Proposition 2.27. Q_{A}Cess^{∗}(BG) = H_{𝔪}^{0,∗}(Q_{A}Cess^{∗}(BG)) = H_{𝔪}^{0,∗}(Q_{A}H^{∗}(BG)).

Our argument is similar to that proving [K3, Prop.8.9]. We need

Lemma 2.28 ([K3, Lem.8.8]). Assume c < r. Given any sequence z_{1}, …, z_{c} ∈ H^{∗}(G) that generates the polynomial algebra A, there exists z ∈ H^{∗}(BG) such that, for all proper inclusions C < V, z_{1}, …, z_{c}, z restricts to a regular sequence in H^{∗}(BC_{G}(V)).

Proof of Proposition 2.27. As Q_{A}Cess^{∗}(BG) is finite dimensional, we clearly have

Q_{A}Cess^{∗}(BG) = H_{𝔪}^{0,∗}(Q_{A}Cess^{∗}(BG)).

By Proposition 2.18, we have an exact sequence

0 → Q_{A}Cess^{∗}(BG) → Q_{A}H^{∗}(BG) → ∏_{C(G)≨V} Q_{A}H^{∗}(BC_{G}(V)),

and this induces an exact sequence

0 → H_{𝔪}^{0,∗}(Q_{A}Cess^{∗}(BG)) → H_{𝔪}^{0,∗}(Q_{A}H^{∗}(BG)) → ∏_{C(G)≨V} H_{𝔪}^{0,∗}(Q_{A}H^{∗}(BC_{G}(V))).

But the last term here is 0, because if z ∈ H^{∗}(BG) is chosen as in the lemma, then z will act regularly on each Q_{A}H^{∗}(BC_{G}(V)) with C(G) ≨ V. □

The last two propositions combine to prove the next theorem.

Theorem 2.29. e_{indec}(G) = e(G) + max{e | H_{𝔪}^{c(G),−c(G)+e}(H^{∗}(BG)) ≠ 0}.

Proof of Theorem 2.22. Symonds [Sy] has proved that

H_{𝔪}^{s,t}(H^{∗}(BG)) = 0 if s + t > − dim G.

Combined with the last theorem, this immediately implies the first part of the theorem: for all compact Lie group G,

e_{indec}(G) ≤ e(G) − dim G.

Furthermore, this inequality will be strict if and only if

H_{𝔪}^{c(G),−c(G)−dim(G)}(H^{∗}(BG)) = 0.

To deduce more, we need to recall why Symond’s result (in the finite group case) had been conjectured by Benson. As constructed by J.P.C.Greenlees and Benson [BG], there is a spectral sequence

H_{𝔪}^{s,t}(H^{∗}(BG)) = E_{2}^{s,t} ⇒ H̃_{−s−t}(EG_{+} ∧_{G} S^{Ad(G)}; 𝔽_{p}),

where S^{Ad(G)} is the one point compactification of the adjoint representation, so Benson was conjecturing that some evident vanishing at the level of E_{∞} happened already at E_{2}.

By Symonds’ theorem, the group H_{𝔪}^{c(G),−c(G)−dim(G)}(H^{∗}(BG)) consists of permanent cycles, as the differentials will take values in groups that are zero. As this group is certain not in the image of nonzero boundary maps, it will thus be a quotient of

H̃_{dim(G)}(EG_{+} ∧_{G} S^{Ad(G)}; 𝔽_{p}) ≃ \begin{cases}𝔽_{p} & if Ad(G) is 𝔽_{p}–oriented. \\ 0 & if not. \end{cases}

In the oriented case, H_{𝔪}^{r(G),−r(G)−dim(G)}(H^{∗}(BG)) ≃ 𝔽_{p}, by a generalization to all compact Lie groups of Benson’s argument [B1] in the finite group case. (The generalization is straightforward, using the transfer map H^{∗}(BV) → H^{∗+dim(G)}(BG) associated to an inclusion V < G.)

Thus, in either the oriented or nonoriented case, we see that

H_{𝔪}^{c(G),−c(G)−dim(G)}(H^{∗}(BG)) = 0

unless c(P) = r(P), i.e. G is p–central. In the p–central case, G will be oriented and e_{indec}(G) = e(G). But arguing as in [K3], one can do better: the top class in Q_{A}H^{∗}(BG) will be represented by a H^{∗}(BC)–primitive, so e_{prim}(G) = e_{indec}(G).

□

We end this section by noting that our results above include a proof of Carlson’s Depth Conjecture in the case of minimal depth, generalizing results in [G, K3]. Note that

e_{indec}(G) ≠ −∞ ⇔ e_{indec}(G) ≥ 0 ⇔ Q_{A}Cess^{∗}(G) ≠ 0 ⇔ Cess^{∗}(G) ≠ 0,

and

H_{𝔪}^{c(G),∗}(H^{∗}(BG)) ≠ 0 ⇔ H^{∗}(BG) has depth precisely c(G).

Therefore, Theorem 2.29 tells us most of the following, and Symond’s theorem tells us the rest.

Theorem 2.30. For G compact Lie, H^{∗}(BG) has depth precisely c(G) if and only if H^{∗}(BG) is not detected by restriction to the cohomology rings H^{∗}(BC_{G}(V)) for V < G of rank greater than c(G). In this case, H_{𝔪}^{c(G),t}(H^{∗}(BG)) ≠ 0 for some −c(G) − dim(G) ≤ t ≤ −c(G) − e(G).

Corollary 2.31. If G is compact Lie, and e(G) < dim(G) then H^{∗}(BG) has depth greater than c(G) and is detected by restriction to the cohomology rings H^{∗}(BC_{G}(V)) for V < G of rank greater than c(G).

3. New results for finite p groups

We now prove various new results about e(P) when P is a finite p–group. We begin with a proof of Theorem 1.7, with part of the discussion relevant for all compact Lie groups G. We will next deduce Theorem 1.5 and Theorem 1.6 assuming Theorem 1.9. Finally we will prove Theorem 1.9, which will involve an excursion into invariant theory.

3.1. Upper bounds for e(P) coming from Chern classes. We use Chern classes of representation to get group theoretic upper bounds for e(P) when P is a finite p–group. With C = C(P), we need to get a lower bound on im(i^{∗}), the image of restriction

i^{∗} : H^{∗}(BP) → H^{∗}(BC).

To set up notation and unify exposition, let c = c(P) and let

H^{∗}(BC) ≃ \begin{cases}𝔽_{2}[x_{1}, …, x_{c}] & if p = 2 \\ Λ(x_{1}, …, x_{c}) ⊗ 𝔽_{p}[y_{1}, …, y_{c}] & if p is odd, \end{cases}

where y_{i} ∈ H^{2}(BC) denotes β(x_{i}) for all primes (so that y_{i} = x_{i}^{2} when p = 2). Note that each element y_{i} is the Chern class of a unique one dimensional complex representation ω_{i} of C.

Now let A < P be a maximal abelian subgroup, so that A certainly contains C. Each ω_{i} extends, possibly nonuniquely, to a one dimensional representation ω̃_{i} of A. Now let ρ_{i} = Ind_{A}^{P}(ω̃_{i}), a representation of P of dimension [P : A] = |P |/|A|.

By construction, the restriction of ρ_{i} to C will be |P |/|A|ω_{i}, which has top Chern class y_{i}^{|P |/|A|}. We have proved the next theorem, a precise form of Theorem 1.7.

Theorem 3.1. The Hopf algebra im(i^{∗}) contains 𝔽_{p}[y_{1}^{|P |/|A|}, …, y_{c}^{|P |/|A|}]. Thus e(P) ≤ c(2|P |/|A| − 1).

Remark 3.2. Let e_{grp}(P) = c(P)(2|P |/|A|−1), where A < P is an abelian subgroup of maximal order; thus the theorem says that e(P) ≤ e_{grp}(P). With arguments similar, but simpler, to ones we will use in the proof of Theorem 1.9, it is not hard to prove that this invariant of p–groups has the following monotonicity property:

if Q < P, then e_{grp}(Q) ≤ e_{grp}(P).

This property suffices to deduce that if P is a finite p–group, then d^{U}(P) ≤ e_{grp}(P):

d^{U}(P) ≤ max_{V <G}{e(C_{G}(V))} ≤ max_{V <G}{e_{grp}(C_{G}(V))} ≤ e_{grp}(P).

3.2. Conjectural upper bounds for e(G) coming from Chern classes. We continue in the spirit of the last subsection, and discuss how one might use Chern classes to prove Conjecture 1.10. This says that, if n(G) is the minimal dimension of a faithful representation of a compact Lie group G, then e(G) ≤ 2n(G) − c(G).

With C = C(G), the calculation of e(G) requires understanding of the Hopf algebra im(i^{∗}), the image of the restriction

i^{∗} : H^{∗}(BG) → H^{∗}(BC).

Definitions 3.3. (a) If ρ is a representation of C, let H(ρ) ⊂ H^{∗}(BC) be the smallest Hopf algebra containing its Chern classes. When ρ is faithful, H^{∗}(BC) will be a finitely generated H(ρ) module, and, in this case, let e(ρ) be the top degree of Q_{H(ρ)}H^{∗}(BC).

(b) If G is a compact Lie group with C = C(G), let H(G) be the smallest Hopf algebra containing all of the H(ρ), where ρ ranges over all representations of G, restricted to C, and let e_{rep}(G) be the top degree of Q_{H(G)}H^{∗}(BC).

It is clear that for any representation ρ of G,

H(ρ) ⊆ H(G) ⊆ im(i^{∗}),

so we learn the following.

Proposition 3.4. e(G) ≤ e_{rep}(G) ≤ e(ρ).

Thus Conjecture 1.10 would follow immediately from the next conjecture, which just concerns Chern classes of representations of elementary abelian groups.

Conjecture 3.5. Let C be an elementary abelian p–group of rank c. If ρ is a faithful n dimensional complex representation of C, then e(ρ) ≤ 2n − c.

In turn, this conjecture would be consequence of a conjectural identification of the Hopf algebra H(ρ). To describe this, it is convenient to use more basis free notation.

Let V = ⟨y_{1}, …, y_{c}⟩, so that H(ρ) ⊂ S^{∗}(V) = 𝔽_{p}[y_{1}, …, y_{c}] ⊂ H^{∗}(BC). Given a subspace W < V, we let Φ^{i}(W) ⊂ S^{p^{i}}(V) be the span of the p^{i}th powers of the elements in W. General sub Hopf algebras of S^{∗}(V) then correspond to filtrations of V as follows.

Definition 3.6. Let F be a finite length filtration of the 𝔽_{p}–vector space V:

V (0) ⊆ V (1) ⊆ ⋯ ⊆ V (k) = V.

The corresponding Hopf algebra H(F) is then defined to be

H(F) = S^{∗}(V (0) + Φ(V (1)) + ⋯ + Φ^{k}(V (k))).

Note that e(F), the top degree of Q_{H(ρ)}(H^{∗}(BC)), can be computed as follows. Let c_{i}(F) be the rank of V (i)/V (i − 1), so that c_{0}(F) + ⋯ + c_{k}(F) = c. Then

e(F) = 2 (∑_{i=0}^{k} c_{i}(F)p^{i}) − c.

Now suppose ρ is a faithful n dimensional complex representation of C. This will be a sum of line bundles, possibly with multiplicities, and so will correspond to the following data:

• A finite set of distinct elements v_{1}, …, v_{m} ∈ V which span V.

• Multiplicities n_{1}, …, n_{m} ∈ ℕ such that n_{1} + ⋯ + n_{m} = n.

From this data, we define a filtration F_{ρ} of V by letting V (i) be the span of the v_{j} such that p^{i+1} does not divide n_{j}.

Lemma 3.7. H(ρ) ⊆ H(F_{ρ}).

Proof. Let ch(ρ) denote the total Chern class. We will have

ch(ρ) = ∏_{j=1}^{m}(1 + v_{j})^{n_{j}} = ∏_{i} ∏_{v_{j}∈V (i)−V (i−1)}(1 + v_{j}^{p^{i}})^{n_{j}/p^{i}}.

As v_{j}^{p^{i}} ∈ H(F_{ρ}) for v_{j} ∈ V (i)−V (i−1), we see that all the homogenous components of ch(ρ) are in H(F_{ρ}) as well. □

We conjecture equality in the last lemma.

Conjecture 3.8. H(ρ) = H(F_{ρ}).

As the estimate e(F_{ρ}) ≤ 2n − c is easily checked, this conjecture implies Conjecture 3.5, and thus Conjecture 1.10.

Remark 3.9. Note that, for any V < G, n(C_{G}(V)) ≤ n(G) and c(C_{G}(V)) ≥ c(G). Thus, if Conjecture 1.10 were true, we could deduce

d^{U}(G) ≤ max_{V <G}{e(C_{G}(V))} ≤ max_{V <G}{2n(C_{G}(V)) − c(C_{G}(V))} ≤ 2n(G) − c(G).

3.3. Proofs of Theorem 1.5 and Theorem 1.6 assuming Theorem 1.9. Here we assume Theorem 1.9, which says that if P is a p-group, and Q < P, then e(Q) ≤ e(P), and deduce Theorem 1.5 and Theorem 1.6.

Proof of Theorem 1.5. This is immediate: d^{U}(P) ≤ max_{V <P}{e(C_{P} (V))} ≤ e(P). □

Proof of Theorem 1.6. Suppose a p–group P acts faithfully on a set S with no fixed points. We wish to show that e(P) ≤ |S|/2 − |S/P | when p = 2, and e(P) ≤ 2|S|/p − |S/P | when p is odd.

Note that S/P is the set of orbits of S, so S has a decomposition into orbits

S = ∐_{i=1}^{|S/P |} S_{i},

with |S_{i}| = p^{r_{i}}, and each r_{i} ≥ 1. Then P admits an embedding

P ⊆ ∏_{i=1}^{|S/P |} W (r_{i}),

where W (r) denotes the Sylow subgroup of the symmetric group Σ_{p^{r}}.

Assuming Theorem 1.9, we would then have the bound

e(P) ≤ ∑_{i=1}^{|S/P |} e(W (r_{i})).

The next proposition will thus complete the proof of Theorem 1.6 □

Proposition 3.10. When p = 2, e(W (r)) = 2^{r−1}−1. When p is odd, e(W (1)) = 0, and, for r ≥ 1, then e(W (r)) = 2p^{r−1} − 1.

Proof. We begin by identifying C(r) = C(W (r)). We claim that C(r) ≃ ℤ/p. This is easily proved by induction on r, as W (r + 1) is the semidirect product

W (r + 1) = W (r)^{p} ⋊ ℤ/p.

so that

C(r + 1) = (C(r)^{p})^{ℤ/p},

the diagonal copy of C(r) in C(r)^{p}.

Now we determine im(i(r)^{∗}) ⊂ H^{∗}(BC(r)), where i(r) : C(r) → W (r) is the inclusion.

The case when r = 1 is elementary: C(1) = W (1) = ℤ/p, so im(i(1)^{∗}) = H^{∗}(Bℤ/p) and e(W (1)) = 0 for all primes p.

To proceed by induction, we observe that the inclusions

C(r + 1) → C(r)^{p} → W (r)^{p} → W (r + 1)

induce a factorization of i(r + 1)^{∗} as

H^{∗}(BW (r + 1)) ↠ H^{∗}(BW (r)^{p})^{ℤ/p} ⟶^{(i(r)^{p})^{∗}} H^{∗}(BC(r)^{p})^{ℤ/p} → H^{∗}(BC(r + 1)),

with the first map epic as indicated.

Now let p be odd. Identifying H^{∗}(BC(r)) with Λ(x) ⊗ 𝔽_{p}[y], we prove by induction that, for r ≥ 2, im(i(r)^{∗}) = 𝔽_{p}[y^{p^{r−1}}] so that e(W (r)) = 2p^{r−1} − 1.

The case when r = 2 is slightly special: im(i(2)^{∗}) will be the image of

(Λ(x_{1}, …, x_{p}) ⊗ 𝔽_{p}[y_{1}, …, y_{p}])^{ℤ/p} → Λ(x) ⊗ 𝔽_{p}[y]

under the map induced by sending each x_{i} to x and y_{i} to y. Recall also that this image will be a Hopf algebra. As y^{p} is the image of the invariant y_{1} ⋯ y_{p}, while x and y are easily checked to not be in this image, we see that im(i(2)^{∗}) = 𝔽_{p}[y^{p}].

Assume by induction that im(i(r)^{∗}) = 𝔽_{p}[y^{p^{r−1}}]. Then, reasoning as above,

im(i(r + 1)^{∗}) = im{𝔽_{p}[y_{1}^{p^{r−1}}, …, y_{p}^{p^{r−1}}]^{ℤ/p} → 𝔽_{p}[y]} = 𝔽_{p}[y^{p^{r}}].

The case when p = 2 is similar. Identifying H^{∗}(BC(r)) with 𝔽_{2}[x], one proves by induction that, for r ≥ 1, im(i(r)^{∗}) = 𝔽_{2}[x^{2^{r−1}}] so that e(W (r)) = 2^{r−1} − 1.

□

3.4. Reduction of Theorem 1.9 to invariant theory. We begin the proof of Theorem 1.9. Our goal is to show that, if Q is a subgroup of a p–group P, then e(Q) ≤ e(P). Thus we need to somehow compare the image of the restriction

H^{∗}(BP) → H^{∗}(BC(P))

to the image of the restriction

H^{∗}(BQ) → H^{∗}(BC(Q)).

We make some first reductions.

First of all, by induction of the index of Q in P, we can assume that Q has index p, and thus will be normal in P. Then ℤ/p ≃ P/Q will act on H^{∗}(BQ) and also on C(Q), with C(Q)^{ℤ/p} = C(P) ∩ Q.

Next, suppose that C(P) is not contained in Q. Then there would exist a central element σ ∈ P of order p, not in Q. It follows easily that then ⟨σ⟩ × Q = P, and we conclude that e(P) = e(Q).

Thus we will assume that C(P) is contained in Q. Suppose P admits a direct product decomposition P = ⟨σ⟩×P_{1}, with σ of order p. Then σ would be contained in C(P) and thus Q = ⟨σ⟩ × Q_{1} with Q_{1} = P_{1} ∩ Q. Then e(P) = e(P_{1}) and e(Q) = e(Q_{1}).

We are reduced to needing to prove that e(Q) ≤ e(P) under the following assumptions:

• Q is normal of index p, so ℤ/p ≃ P/Q acts on both H^{∗}(BQ) and C = C(Q).

• C(P) = C^{ℤ/p}.

• P has no nontrivial elementary abelian direct summands.

In this situation, the restriction map H^{∗}(BP) → H^{∗}(C(P)) factors

H^{∗}(BP) → H^{∗}(BQ)^{ℤ/p} → H^{∗}(BC)^{ℤ/p} ↪ H^{∗}(BC) ↠ H^{∗}(BC^{ℤ/p}),

and the last assumption tell us that the image lands in the part of H^{∗}(BC^{ℤ/p}) generated by β(H^{1}(BC^{ℤ/p})).

Let V denote β(H^{1}(BC)) ⊆ H^{2}(BC), so V is just the dual of the vector space C. Let V_{ℤ/p} denote the ℤ/p–coinvariants V/⟨x − σx : x ∈ V ⟩, where σ generates ℤ/p, so that the part of H^{∗}(BC^{ℤ/p}) generated by β(H^{1}(BC^{ℤ/p})) identifies with S^{∗}(V_{ℤ/p}).

The image of H^{∗}(BQ) → H^{∗}(BC) ↠ S^{∗}(V) will be the Hopf algebra H(F) associated to a filtration F of V:

V (0) ⊆ V (1) ⊆ ⋯ ⊆ V (n) = V.

Here we are using notation as in §3.2: given W < V, Φ^{k}(W) ⊂ S^{p^{k}}(V) denotes the span of the p^{k}th powers of the elements in W, and then

H(F) = S^{∗}(V (0) + Φ(V (1)) + ⋯ + Φ^{n}(V (n))).

It is important to note that the filtration F will be preserved by the ℤ/p action on V. One way to see this is to note that F is a natural invariant of Q: as described in [K3, §6], this filtration of V records the history of the 0–line in the spectral sequence associated to the central extension

C → Q → Q/C.

From our observations above, the image of H^{∗}(BP) → H^{∗}(C(P)) will be contained in the image of

H(F)^{ℤ/p} ↪ S^{∗}(V)^{ℤ/p} ↪ S^{∗}(V) ↠ S^{∗}(V_{ℤ/p}).

Recall that e(F) is defined to be ∑_{k=0}^{n} c_{k}(F)(2p^{k} − 1) where c_{k}(F) is the rank of

V (k)/V (k − 1). As e(Q) ≤ e(F), we will be able to deduce that e(Q) ≤ e(P) if we can solve the following problem in invariant theory.

Problem 3.11. Given a filtration F of a ℤ/p–module V, find a filtration F_{ℤ/p} of V_{ℤ/p} such that

• The image of H(F)^{ℤ/p} → S^{∗}(V_{ℤ/p}) is contained in H(F_{ℤ/p}), and

• e(F) ≤ e(F_{ℤ/p}).

In the next section we find such a filtration F_{ℤ/p}: see Theorem 4.6.

4. New results in invariant theory

In this section F is a filtration of an 𝔽_{p}[ℤ/p]–module V,

V (0) ⊆ V (1) ⊆ ⋯ ⊆ V (n) = V,

and we wish to understand the image of the composite

H(F)^{ℤ/p} ↪ S^{∗}(V)^{ℤ/p} ↪ S^{∗}(V) ↠ S^{∗}(V_{ℤ/p}),

with our goal to solve Problem 3.11. Throughout we let σ be a generator for ℤ/p.

4.1. ℤ/p–modules. The modular representation theory of ℤ/p is quite tame. There are p indecomposable 𝔽_{p}[ℤ/p]–modules, V_{1}, …, V_{p}, where V_{i} has dimension i. An explicit model for V_{i} is the vector space with basis x_{1}, …, x_{i} with

σx_{j} = \begin{cases}x_{j} + x_{j−1} & if 1 < j ≤ i \\ x_{1} & if j = 1. \end{cases}

A general 𝔽_{p}[ℤ/p]–module V decomposes as a direct sum

V ≃ m_{1}V_{1} ⊕ m_{2}V_{2} ⊕ ⋯ ⊕ m_{p}V_{p}.

We say that V is reduced if m_{1} = 0.

We let rad(V) and soc(V) be the radical and socle of a module V. Thus soc(V) = V^{ℤ/p} and V/rad(V) = V_{ℤ/p}. In the usual way, we define soc(V) ⊂ soc^{2}(V) ⊂ … and rad(V) ⊃ rad^{2}(V) ⊃ . . ..

The submodule m_{1}V_{1} in a decomposition of V can be regarded as the image of a section of the quotient map soc(V) ↠ soc(V)/soc(V) ∩ rad(V). Thus V is reduced precisely when soc(V) ⊂ rad(V), or equivalently, when the composite V^{ℤ/p} ↪ V ↠ V_{ℤ/p} is zero.

4.2. The case when the filtration is trivial. Given a ℤ/p–module V, a special case of our general problem is to understand the image of

S^{∗}(V)^{ℤ/p} ↪ S^{∗}(V) ↠ S^{∗}(V_{ℤ/p}).

We remark that, in spite of the simple classification of modules V, a complete calculation of S^{∗}(V)^{ℤ/p} is not known in all cases, and is the subject of much research. Even so, we prove the following theorem.

Theorem 4.1. If V = W ⊕ U, where W is trivial and U is reduced, the image of S^{∗}(V)^{ℤ/p} → S^{∗}(V_{ℤ/p}) is S^{∗}(W ⊕ Φ(U_{ℤ/p})).

Here is a more invariant way of stating this. Given V, let W_{ℤ/p} be the image of the composite V^{ℤ/p} ↪ V ↠ V_{ℤ/p}. Then the image of

S^{∗}(V)^{ℤ/p} ↪ S^{∗}(V) ↠ S^{∗}(V_{ℤ/p})

will be

S^{∗}(W_{ℤ/p} + Φ(V_{ℤ/p})).

The next example both illustrates the theorem and will be used in its proof.

Example 4.2. Suppose V = mV_{2}, where the ith copy of V_{2} has basis {x_{i}, y_{i}} with σy_{i} = y_{i} + x_{i} and σx_{i} = x_{i}. The kernel of the quotient V ↠ V_{ℤ/p} is the span of the x_{i}’s, so we can view V_{ℤ/p} as having basis given by the y_{i}’s. The theorem in this case is asserting that the image of the composite

𝔽_{p}[x_{1}, …, x_{m}, y_{1}, …, y_{m}]^{ℤ/p} ↪ 𝔽_{p}[x_{1}, …, x_{m}, y_{1}, …, y_{m}] ↠ 𝔽_{p}[y_{1}, …, y_{m}]

is 𝔽_{p}[y_{1}^{p}, …, y_{m}^{p}]. The main theorem of [CH] is a description of generators of S^{∗}(mV_{2})^{ℤ/p} as polynomials in the x_{i}’s and y_{j}’s; see also [W]. One sees that all of these are sent to 0 modulo the ideal (x_{1}, …, x_{m}) except for the ‘norm’ generators ∏_{j=0}^{p−1} σ^{j}y_{i} = y_{i}^{p} − x_{i}^{p−1}y_{i}, which map to y_{i}^{p}. So the assertion of the theorem is true in this case.

Proof of Theorem 4.1. First we note that if V = W ⊕ U with W trivial, then S^{∗}(V)^{ℤ/p} = S^{∗}(W) ⊗ S^{∗}(U)^{ℤ/p} and S^{∗}(W ⊕ Φ(U_{ℤ/p})) = S^{∗}(W) ⊗ S^{∗}(Φ(U_{ℤ/p})). So it suffices to prove that, when V is reduced, there is an equality

I(V) = S^{∗}(Φ(V_{ℤ/p})),

where I(V) = im{S^{∗}(V)^{ℤ/p} ↪ S^{∗}(V) ↠ S^{∗}(V_{ℤ/p})}.

The previous example showed that this holds when V = mV_{2}. We use this to show that the equality holds for a general reduced V. Recall that V_{ℤ/p} = V/rad(V). If we let V̄ = V/rad^{2}(V), and let Ṽ be the projective cover of V, then V̄ = mV_{2} and Ṽ = mV_{p}, where V_{ℤ/p} = mV_{1}. The surjections Ṽ ↠ V ↠ V̄ will induce isomorphisms Ṽ_{ℤ/p} = V_{ℤ/p} = V̄_{ℤ/p}, and then inclusions

I(Ṽ) ⊆ I(V) ⊆ I(V̄) = S^{∗}(Φ(V_{ℤ/p})).

Finally, to see that all of these inclusions are, in fact, equalities, we note I(Ṽ_{p}) is easily seen to contain S^{∗}(Φ(V_{ℤ/p})): our proof of Proposition 3.10 showed that I(V_{p}) = S^{∗}(Φ(V_{1})), and so I(mV_{p}) certainly contains S^{∗}(Φ(mV_{1})). □

4.3. The case when V = W ⊕U with W trivial and U reduced. Now suppose that there exists a decomposition of filtered ℤ/p–modules V = W ⊕U, with W trivial and U reduced. Define a filtration F_{ℤ/p} of V_{ℤ/p} by letting

V_{ℤ/p}(k) = (W (k) + U(k − 1) + rad(V))/rad(V).

Proposition 4.3. The image of H(F)^{ℤ/p} → S^{∗}(V_{ℤ/p}) is contained in H(F_{ℤ/p}).

Proof. Just as in the proof of Theorem 4.1, it suffices to prove this when V is reduced, so that F_{ℤ/p} is defined by letting V_{ℤ/p}(k) = (V (k − 1) + rad(V))/rad(V).

Also, similar to the proof of Theorem 4.1, we let V̄ = V/rad^{2}(V), with filtration F̄ defined by V̄ (k) = (V (k) + rad^{2}(V))/rad^{2}(V). Then

im{H(F)^{ℤ/p} → S^{∗}(V_{ℤ/p})} ⊆ im{H(F̄)^{ℤ/p} → S^{∗}(V_{ℤ/p})},

and the filtrations F_{ℤ/p} and F̄_{ℤ/p} of V_{ℤ/p} agree.

Thus it suffices to also assume that V satisfies rad^{2}(V) = 0. In this case, let elements y_{1}, …, y_{t} ∈ V, of filtration k_{1}, …, k_{t}, project to a filtered basis of V_{ℤ/p}, and let x_{j} = σy_{j}. Then

H(F) ⊆ 𝔽_{p}[x_{1}, …, x_{t}, y^{p^{k_{1}}} , …, y^{p^{k_{t}}}]

as algebras with ℤ/p action, and so the image of H(F)^{ℤ/p} → S^{∗}(V_{ℤ/p}) is contained in H(F_{ℤ/p}) = 𝔽_{p}[y^{p^{k_{1}} +1}, …, y^{p^{k_{t}} +1}], as this is the image of

𝔽_{p}[x_{1}, …, x_{t}, y^{p^{k_{1}}} , …, y^{p^{k_{t}}}]^{ℤ/p} → 𝔽_{p}[y_{1}, …, y_{t}].
□

4.4. The general case. Unfortunately, at least when p ≥ 3, a general filtered ℤ/p–module V need not admit a direct sum decomposition as filtered modules of the form V = W ⊕ U, with W trivial and U reduced.

Example 4.4. With p ≥ 3, let V (0) = V_{2} embedding ‘diagonally’ in V_{1} ⊕ V_{3} = V (1) = V. Then the image of soc(V) → V/rad(V) is V_{1}, generated by an element of V (0), but not of soc(V (0)), and we see that there is no isomorphism V ≃ V_{1} ⊕V_{3} as filtered modules^{1}.

This phenomenon goes away if we assume that rad(V) ⊆ V (0).

Lemma 4.5. If rad(V) ⊆ V (0), then there exists a decomposition of filtered ℤ/p– modules V = W ⊕ U, with W trivial and U reduced.

We temporarily postpone the proof.

Now let F be an arbitrary filtration of a ℤ/p–module V. Define a filtration F_{ℤ/p} of V_{ℤ/p} by letting

V_{ℤ/p}(k) = (soc(V (k) + rad(V)) + V (k − 1) + rad(V))/rad(V).

Note that, if V = W ⊕ U with W trivial and U reduced, then the filtration F_{ℤ/p} agrees with the filtration of the same name in the last subsection.

Theorem 4.6. The image of H(F)^{ℤ/p} → S^{∗}(V_{ℤ/p}) is contained in H(F_{ℤ/p}).

Proposition 4.7. e(F) ≤ e(F_{ℤ/p}).

Proof of Theorem 4.6. Let F′ be the filtration of V defined by letting

V′(k) = V (k) + rad(V).

Then H(F) ⊆ H(F′), so that

im{H(F)^{ℤ/p} → S^{∗}(V_{ℤ/p})} ⊆ im{H(F′)^{ℤ/p} → S^{∗}(V_{ℤ/p})}.

Moreover, F′_{ℤ/p} = F_{ℤ/p}. Thus the proposition follows from Proposition 4.3. □

Proof of Proposition 4.7. One can choose a filtered basis for V consisting of elements x_{α}, y_{β}, z_{γ} such that the set of elements z̄_{γ} = z_{γ} + rad(V) span (soc(V) + rad(V))/rad(V), and the set of elements x_{α} span rad(V). Each y_{β} will generate a ℤ/p–submodule of dimension at most p, and with radical of filtration at most equal to |y_{β}|, the filtration of y_{β}, and such modules, together with the z_{γ}, span V. In V_{ℤ/p}, the element ȳ_{β} = y_{β} + rad(V) will have filtration |y_{β} + 1|, while |z̄_{β}| ≥ |z_{β}|, with the possibility of > due to the phenomenon illustrated in Example 4.4.

It follows that

e(F) = ∑_{α} 2p^{|x_{α}|} + ∑_{β} 2p^{|y_{β}|} + ∑_{γ} 2p^{|z_{γ}|} − r(V)

≤ ∑_{β} 2p^{|y_{β}|+1} + ∑_{γ} 2p^{|z_{γ}|} − r(V)

≤ ∑_{β} 2p^{|ȳ_{β}|} + ∑_{γ} 2p^{|z̄_{γ}|} − r(V_{ℤ/p})

= e(F_{ℤ/p}).

□

Proof of Lemma 4.5. Filter V_{ℤ/p} by letting F_{k}V_{ℤ/p} = (V (k) + rad(V))/rad(V). Then let W_{ℤ/p} = (soc(V)+rad(V))/rad(V) ⊂ V_{ℤ/p} be filtered by letting F_{k}W_{ℤ/p} = W_{ℤ/p} ∩ F_{k}V_{ℤ/p}. It is easy to choose a filtered complement U_{ℤ/p} so that F_{k}V_{ℤ/p} = F_{k}W_{ℤ/p} ⊕ F_{k}U_{ℤ/p} as filtered ℤ/p–vector spaces.

1We thank Dave Benson for showing us this example.

The point is now that, as rad(V) ⊆ V (0), one can choose a lifting

\xymatrix{ & V \ar[d] \\ W_{ℤ/p} \ar[r] \ar@{.>}[ur] & V_{ℤ/p}}

as filtered vector spaces so that the image is contained in soc(V), and thus can be viewed as a lifting of filtered ℤ/p–modules. For if x + rad(V) = y + rad(V) with x ∈ V (k) and y ∈ soc(V), then y ∈ V (k) ∩ soc(V) = soc(V (k)). The conclusion of the lemma follows if we let W be the image of such a lifting, and U equal to the filtered ℤ/p–module generated by any lifting

\xymatrix{ & V \ar[d] \\ U_{ℤ/p} \ar[r] \ar@{.>}[ur] & V_{ℤ/p}.}

□

Example 4.8. We illustrate how Theorem 4.6 and Proposition 4.7 work when our filtered module V is as in Example 4.4. Thus let p ≥ 3, and let F be the filtration given by having V (0) = V_{2} ‘diagonally’ embedded in V = V (1) = V_{1} ⊕ V_{3}. Then F_{ℤ/p} is the filtration having V_{ℤ/p}(0) = V_{1} embedded as the first factor of V_{ℤ/p} = V_{ℤ/p}(1) = V_{1} ⊕ V_{1}.

Explicitly, V = V (1) = ⟨x_{1}, x_{2}, y_{1}, y_{2}⟩, and V (0) = ⟨x_{1}, x_{2}⟩, with action: σ(y_{2}) = y_{2} + y_{1} + x_{2}, σ(y_{1}) = y_{1} + x_{1}, σ(x_{2}) = x_{2} + x_{1}, σ(x_{1}) = x_{1}. The quotient map V → V_{ℤ/p} identifies with the map V → ⟨x̄_{2}, ȳ_{2}⟩ sending x_{1} to 0, x_{2} and −y_{1} to x̄_{2}, and y_{2} to ȳ_{2}. Then Theorem 4.6 says that the image of 𝔽_{p}[x_{1}, x_{2}, y_{1}^{p}, y_{2}^{p}]^{ℤ/p} in 𝔽_{p}[x̄_{2}, ȳ_{2}] will be contained in 𝔽_{p}[x̄_{2}, ȳ_{2}^{p^{2}}]. Note that e(F) = 4p+2 ≤ 2p^{2} = e(F_{ℤ/p}), as predicted by Proposition 4.7.

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Department of Mathematics, University of Virginia, Charlottesville, VA 22904

E-mail address: njk4x@virginia.edu