The action of the Steenrod algebra on Tate cohomology

D. J. Benson Mathematical Institute 24–29 St. Giles Oxford OX1 3LB Great Britain

J. P. C. Greenlees Department of Mathematics Hicks Building Sheffield S10 2QN Great Britain

Abstract

Let G be a finite group and k a field of characteristic p dividing |G|. In this paper, we examine the action of the Steenrod operations on the Tate cohomology Ĥ^{∗}(G, k). In particular, we prove that all Steenrod operations from negative to positive degree Tate cohomology vanish if and only if all products in negative degree Tate cohomology vanish.

1 Introduction

Let G be a finite group and k a field of characteristic p dividing |G|. A recent paper [1] examined the question of when there can be non-zero products between elements of negative degree in Tate cohomology Ĥ^{−}(G, k). It was observed that if there is such a non-zero product, then there is an element y ∈ H^{∗}(G, k) of positive degree, such that the Krull dimension of H^{∗}(G, k)/Ann(y) is equal to one.

Our aim in this paper is to relate the above question to the action of the Steenrod algebra A on Tate cohomology. Namely, we have a short exact sequence

0 → H^{∗}(G, F_{p}) → Ĥ^{∗}(G, F_{p}) → Ĥ^{−}(G, F_{p}) → 0.

The statement that this splits as a sequence of H^{∗}(G, F_{p})-modules is equivalent to the statement that all products on Ĥ^{−}(G, F_{p}) are zero ([1], Lemma 2.1). On the other hand, the statement that this splits as a sequence of A-modules is equivalent to the statement that there are no non-zero Steenrod operations going from negative to positive degree. Our main theorem states that the above sequence splits as H^{∗}(G, F_{p})-modules if and only if it splits as A-modules.

Theorem 1.1 Let G be a finite group and p a prime dividing |G|. Then the following are equivalent:

(i) All products of elements of negative degree Tate cohomology Ĥ^{−}(G, F_{p}) vanish. (ii) All Steenrod operations from negative degree to positive degree on Ĥ^{∗}(G, F_{p}) are

zero.

Corollary 1.2 If there is a non-zero Steenrod operation from negative degree to positive degree on Ĥ^{∗}(G, F_{p}), then there is an element y ∈ H^{∗}(G, F_{p}) such that the Krull dimension of H^{∗}(G, k)/Ann(y) is equal to one.

Corollary 1.3 If the centre of a Sylow p-subgroup of G has rank at least two, then all Steenrod operations from negative degree to positive degree on Ĥ^{∗}(G, F_{p}) are zero.

Corollary 1.4 Suppose that G is a finite group of p-rank two. Then H^{∗}(G, F_{p}) is Cohen– Macaulay if and only if all Steenrod operations from negative degree to positive degree on Ĥ^{∗}(G, F_{p}) are zero.

We end the introduction with a few remarks. First, we remark that in the case where G is cyclic of order p, all Steenrod reduced powers are non-zero on a class of degree −1, so that the extension of A-modules is really very far from split in this case.

The next remark is that in general, one can deduce the action of A on Tate cohomlogy from the action on ordinary cohomology as follows. We have noted that we only have an extension problem, so it remains to calculate the effect of P^{n} when it goes from negative to positive degrees. Specifically, if x ∈ Ĥ^{−}(G, F_{p}) and P^{n}(x) ∈ Ĥ^{+}(G, F_{p}), we choose a non zero divisor ζ ∈ H^{∗}(G, F_{p}) (Lemma 3.1) and a value of r large enough so that ζ^{r}x ∈ Ĥ^{+}(G, F_{p}). We may suppose be induction on n that P^{i}x is known for i < n, and the the Cartan formula gives P^{n}(ζ^{r}x) = ζ^{r}P^{n}(x) + lower terms which are known by induction. Since multiplication by ζ^{r} is injective in positive degrees, this determines P^{n}(x). The proof of the main theorem is a development of this idea.

Finally, we remark that in the case where G is elementary abelian of rank bigger than one, Corollary 1.2 shows that the Steenrod operations are all trivial from negative to positive degrees, which answers a question of J. Morava.

2 Steenrod operations

We begin with a brief discussion of how the Steenrod algebra A acts on Tate cohomology.

It seems most efficient to use a little equivariant topology. Specifically, we shall use the fact that there is a good homotopy category of G-spectra [5] in which equivariant cohomology theories F_{G}^{∗}(•) are represented, in the sense that there is a G-spectrum F such that for any based G-space X we have F_{G}^{∗}(X) = [X, F]_{G}^{∗}. Thus ordinary cohomology of the Borel construction H^{∗}(EG ×_{G} X; F_{p}) is represented by a spectrum b; in particular the cohomology of a point is the group cohomology b^{∗} = H^{∗}(G, F_{p}). The relevance of this is that Tate–Swan cohomology of G-spaces can be represented by the spectrum t = b ∧ ẼG, where ẼG denotes the unreduced suspension of EG [3]; in particular we have t^{∗} = Ĥ^{∗}(G, F_{p}). Now b is a commutative ring spectrum in the sense that there are maps η : S^{0} ⟶ b and μ : b ∧ b ⟶ b which make the unit and associativity diagrams commute; for our purposes the best way to see this is that b may be described as a function spectrum of maps into a well known ring spectrum.

Proposition 2.1 There is an equivalence b = F (EG_{+}, HF_{p}) where HF_{p} is the nonequivariant spectrum representing ordinary mod p cohomology, regarded as a G-spectrum as in Definition II.1.1 of [5].

Proof For a pointed G-space X we have

[X, F (EG_{+}, HF_{p})]_{G}^{∗} = [EG_{+} ∧ X, HF_{p}]_{G}^{∗} = [EG_{+} ∧_{G} X, HF_{p}]^{∗}

where the first equality is formal, and the second is a familiar change of groups isomorphism [5], Theorem II.4.5, valid since EG_{+} is free and HF_{p} is G-fixed. The lemma then follows from uniqueness of representing objects. □

It is then obvious that t is a ring spectrum since ẼG ∧ ẼG ≃ ẼG, and hence it is a b algebra.

Corollary 2.2 The action of A on Tate cohomology satisfies the Cartan formula.

Proof It is familiar that A = [HF_{p}, HF_{p}]^{∗} by the Yoneda lemma, so it follows from Proposition 2.1 that there is a map A ⟶ [b, b]_{G}^{∗}. Indeed in [4] it is shown that with the coefficient ring b^{∗} it gives all stable operations: b^{∗} ⊗ A ≅ [b, b]_{G}^{∗}.

Now, given any operation θ ∈ [b, b]_{G}^{∗} = b^{∗}b we may consider μ^{∗}(θ) ∈ b^{∗}(b ∧ b). Again, it is shown in [4] that b^{∗}(b ∧ b) ≅ b^{∗}b ⊗_{b^{∗}} b^{∗}b, and it follows that there is a diagram

b ∧ b ⟶^{μ} b
∑_{i} θ′_{i}∧θ′′_{i} ↓  ↓ θ
b ∧ b ⟶^{μ} b .

Using this and the fact that t is a b algebra we may construct a diagram which shows for x, y ∈ t^{∗} that θ(xy) = ∑_{i} θ′_{i}(x)θ′′_{i}(y). Using Proposition 2.1 or results from [4] it follows that if θ ∈ A then θ′_{i} and θ′′_{i} are the usual terms in the Cartan formula. □

The other property that we shall need is that an element θ ∈ A should act in dual ways on the positive and negative parts of the Tate cohomology. First we must explain that homology theories are also represented in the category of G-spectra, and in particular the homology of the Borel construction H_{∗}(EG ×_{G} X; F_{p}) is represented by the coBorel spectrum c = b ∧ EG_{+}, which is visibly a b module spectrum. Evidently the coefficient groups of c are the homology groups of G in the sense that c^{−i} = c_{i} = H_{i}(G, F_{p}), and, as might be expected, the action of b^{∗} on c_{∗} corresponds to the action of H^{∗}(G) on H_{∗}(G) by cap product. This may be proved with a little bit of duality theory for finite free G-complexes, using the fact that if X is finite and free D(X/G) = (D_{G}X)/G in an appropriate sense [5], Proposition III.2.12. Furthermore the evident cofibre sequence b ⟶ t ⟶ Σc induces the short exact sequence under consideration as its sequence of homotopy groups; it is obviously a sequence of b^{∗}b modules.

Proposition 2.3 (a) The action of θ on Borel cohomology corresponds to the usual action under the isomorphism b^{∗}(X) = H^{∗}(EG ×_{G} X; F_{p}).

(b) The action of θ on coBorel homology corresponds to the usual action under the isomorphism c_{∗}(X) = H_{∗}(EG ×_{G} X; F_{p}).

Proof Part (a) is clear from Proposition 2.1 and Part (b) follows by duality. □

Corollary 2.4 The action of A on Ĥ^{∗}(G, F_{p}) obeys Tate duality in the sense that if a ≥ 0 the maps θ : Ĥ^{a}(G, F_{p}) ⟶ Ĥ^{a+n}(G, F_{p}) and θ : Ĥ^{−a−n−1}(G, F_{p}) ⟶ Ĥ^{−a−1}(G, F_{p}) are duals.

Proof This follows from Proposition 2.3 since the action of A on homology is defined to be dual to that on cohomology. □

3 Proof of the main theorem

We begin with a lemma describing properties of the ring structure needed in the proof.

Lemma 3.1 (i) (Duflot [2]) H^{∗}(G, F_{p}) has depth at least one. In other words, there is an element ζ of positive degree in Ĥ^{∗}(G, F_{p}) with the property that multiplication by ζ is injective in positive degrees and surjective in negative degrees.

(ii) There exists a pair of elements of Ĥ^{−}(G, F_{p}) whose product is non-zero, if and only if there exists x ∈ Ĥ^{−}(G, F_{p}) with the property that for r > 0 large enough, ζ^{r}x is a non-zero element of positive degree in Ĥ^{∗}(G, F_{p}).

Proof Since (i) is proved in Duflot [2], we shall only prove (ii). By [1], Lemma 2.1, there exists a pair of elements of Ĥ^{−}(G, F_{p}) with non-zero product, if and only if there are elements u ∈ Ĥ^{−}(G, F_{p}) and v ∈ Ĥ^{+}(G, F_{p}) with uv a non-zero element of positive degree. If the latter holds, choose r > 0 large enough so that deg(ζ^{r}) > deg(uv). Since multiplication by ζ^{r} is surjective on Ĥ^{−}(G, F_{p}), we can choose w ∈ Ĥ^{−}(G, F_{p}) with u = ζ^{r}w. Set x = wv, so that deg(x) < 0 by choice of r. Then ζ^{r}x = ζ^{r}wv = uv is a non-zero element of positive degree. □

We begin the proof of the main theorem by assuming that all products in Ĥ^{−}(G, F_{p}) are zero, and proving by induction on n that for x ∈ Ĥ^{−}(G, F_{p}) with deg(P^{n}(x)) ≥ 0, P^{n}(x) = 0 (if p = 2, replace P^{n} by Sq^{n}). Suppose, to the contrary, that P^{n}(x) = y is a non-zero element of positive degree in Ĥ^{∗}(G, F_{p}). Let ζ ∈ H^{∗}(G, F_{p}) be as in part (i) of the lemma, and choose r > 0 large enough so that deg(ζ^{r}) + deg(x) > 0. Since ζ^{r}y ≠ 0, we can choose, by Tate duality, an element z in negative degree so that ζ^{r}yz is non-zero in Ĥ^{−1}(G, F_{p}). Thus yz ≠ 0, but xz = 0. But now the Cartan formula gives

0 = P^{n}(xz) = yz + ∑_{j=1}^{n} P^{n−j}(x)P^{j}(z).

The terms in the sum with deg(P^{n−j}(x)) ≥ 0 vanish by induction, while the remaining terms are products of negative degree elements and so they vanish by hypothesis. This means that yz = 0, and this contradiction completes the induction. We also need to prove that the Bockstein β vanishes on Ĥ^{−1}(G, F_{p}), but of course this happens whenever |G| is divisible by p^{2}.

Conversely, suppose that there is a non-zero product in Ĥ^{−}(G, F_{p}), so that by part (ii) of the lemma, we can choose x ∈ Ĥ^{−m}(G, F_{p}) with xζ^{r} a non-zero element of positive degree in Ĥ^{∗}(G, F_{p}). Let deg(ζ^{r}) = 2n, so that 0 < m < 2n. Note also that ζ is a non-zero divisor in positive degrees, so that xζ^{pr} ≠ 0. Choose z ∈ Ĥ^{−2pn+m−1}(G, F_{p}) with zxζ^{pr} ≠ 0 in Ĥ^{−1}(G, F_{p}). Since P^{n}(ζ^{r}) = ζ^{pr}, it follows by Tate duality that P^{n}(zx) is an element of negative degree whose product with ζ^{r} is non-zero, so that in particular P^{n}(zx) ≠ 0. Now expand this out using the Cartan formula. If 0 ≤ i < m/2p then 2(n − i) > 2pn − m − 2(p − 1)(n − i) and so by the unstable axiom for the action on homology, P^{n−i}(z) = 0. If m/2p ≤ i ≤ m/2(p − 1) then similarly P^{i}(x) = 0. So the only possibility is that P^{i}(x) ≠ 0 for some i > m/2(p − 1), so that deg(P^{i}(x)) > 0. This completes the proof of the converse. We leave to the reader the easy task of rewriting this part of the proof for p = 2.

References

[1] D. J. Benson and J. F. Carlson. Products in negative cohomology. Submitted to J. Pure Appl. Algebra, 1992.

[2] J. Duflot. Depth and equivariant cohomology. Comm. Math. Helvetici 56 (1981), 627– 637.

[3] J.P.C. Greenlees. Representing Tate cohomology of G-spaces. Proc. Edinburgh Math. Soc. 30 (1987) 435-443.

[4] J.P.C. Greenlees. Stable maps into free G-spaces. Trans. American Math. Soc. 310 (1988) 199-215.

[5] L.G. Lewis, J.P. May and M. Steinberger (with contributions by J.E. McClure) Equivariant stable homotopy theory. Lecture notes in mathematics 1213, Springer-Verlag (1986).