On the cohomology of split extensions

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

M. Feshbach Dept. of Mathematics University of Minnesota Minneapolis, MN55455 USA

Abstract

We construct, for each value of n, a split extension of finite 2-groups, with complement isomorphic to Z/2, for which the differential d_{n} is non-zero in the Lyndon– Hochschild–Serre spectral sequence.

AMS Classification 20J06

1 Introduction

In general, the differentials in the Lyndon–Hochschild–Serre spectral sequence for a group extension are difficult to understand. In the case of a central extension, the E_{2} page is easy to calculate, and the transgressions may be computed using the fact that they commute with the Steenrod operations on the base and fiber. Internal differentials, however, are more difficult.

At the other extreme, in the case of a split extension with non-trivial action, even the E_{2} page is hard to calculate. But it appears possible, from the calculations which have been achieved so far, that in the case of a split extension with abelian normal subgroup, one might always have E_{3} = E_{∞}. In case the normal subgroup is a torus (direct product of copies of the circle group S^{1}; in this case it is not really a Lyndon–Hochschild–Serre spectral sequence, but rather the Serre spectral sequence for the corresponding fibration of classifying spaces), the cohomology is concentrated in even degrees, so this would imply that E_{2} = E_{∞}. So the restriction map in this case would be surjective onto the invariant elements of the cohomology of the torus (see [3] for some evidence for this). It also appears possible that if the normal subgroup is an elementary abelian 2-group and the coefficients are taken to be a field of characteristic two, then E_{2} might again always equal E_{∞}.

Our contribution in this paper is to show that in the above speculations, the hypothesis that the normal subgroup is abelian cannot be dropped. Namely we construct, for each value of n, a split extension of finite 2-groups, with complement isomorphic to Z/2, for which the differential d_{n} is non-zero. The groups involved are nilpotent of class two, and the coefficients are taken to be a field of characteristic two. The example is a modification of an example constructed by Avrunin and Carlson [1] to show that the cohomology of a finite group can have nilpotent elements with arbitrarily large degree of nilpotence (multiplicative exponent).

2 Extensions with Z/2 quotient

Suppose that N ⊲ G is a subgroup of index two, and let g be an element of G not in N. Then the extension

1 → N → G → Z/2 → 1

corresponds to an element x ∈ H^{1}(G, F_{2}).

Lemma 2.1 The above extension splits if and only if x is not nilpotent.

Proof If the extension splits then every power of x restricts to a non-zero cohomology class on the complement, and so x is not nilpotent. Conversely, if x is not nilpotent then by a theorem of Quillen [4], x has a non-zero restriction to some elementary abelian 2subgroup, and since it has degree one, it therefore has a non-zero restriction to a cyclic subgroup of order two. This subgroup cannot be contained in N because the restriction of x to N is zero. □

Let k be a field of characteristic two. Then we have a short exact sequence of kGmodules

0 → k → k_{N} ↑^{G}→ k → 0

and hence a long exact sequence in cohomology

⋯ ⟶^{x} H^{n}(G, k) ⟶^{res_{G,N}} H^{n}(N, k) ⟶^{Tr_{N,G}} H^{n}(G, k) ⟶^{x} H^{n+1}(G, k) ⟶^{res_{G,N}} ⋯

Here, we have used the Eckmann–Shapiro isomorphism to identify H^{∗}(G, k_{N} ↑^{G}) with H^{∗}(N, k). The identification of the maps in the above sequence is an easy exercise in diagram chasing. Note that res_{G,N} ∘ Tr_{N,G} is equal to the action of 1 + g on H^{∗}(N, k).

The above long exact sequence is related to the spectral sequence for the group extension as follows. Let P_{∗} be the minimal resolution of k as a k(Z/2)-module, so that each P_{p} is isomorphic to k(Z/2), and let Q_{∗} be a resolution of k as a kG-module. Then the spectral sequence of the group extension may be taken to be the spectral sequence of the double complex

E_{0}^{pq} = Hom_{kG/N}(P_{p}, Hom_{kN}(Q_{q}, k)) ≅ Hom_{kG}(P_{p} ⊗ Q_{q}, k) ≅ Hom_{kN}(Q_{q}, k).

In this spectral sequence, we have

D_{1}^{pq} ≅ D_{1}^{0q} ≅ H^{q}(Tot(E_{0}), d_{0} + d_{1}) ≅ H^{q}(G, k)

(see for example [2], p. 110 and 113). We also have

E_{1}^{pq} = Hom_{kG/N}(P_{p}, H^{q}(N, k)) ≅ H^{q}(N, k),

so that E_{1}^{∗∗} ≅ k[x] ⊗ H^{∗}(N, k), with product given by

(x^{i} ⊗ a)(x^{j} ⊗ b) = x^{i+j} ⊗ (g^{j})^{∗}(a)b.

We have i_{1} = x, j_{1} = res_{G,N} and k_{1} = Tr_{N,G}, so that the differential d_{1} is equal to 1 + g = res_{G,N} ∘ Tr_{N,G}. By the usual identification of the differentials in the spectral sequence (see for example [2], p. 103–105), an element x^{p} ⊗ y survives to represent an element [x^{p} ⊗ y] in E_{n}^{pq} if and only if there is an element z ∈ H^{q−n+1}(G, k) with Tr_{N,G}(y) = x^{n−1}z, and then we have

d_{n}[x^{p} ⊗ y] = [x^{p+n} ⊗ res_{G,N}(z)] ∈ E_{n}^{p+n,q−n+1}.

Unfiltering, it follows that d_{n} ≠ 0 if and only if there is an element z ∈ H^{∗}(G, k) with x^{n}z = 0, but such that there is no element u ∈ H^{∗}(G, k) with x^{n−1}z = x^{n}u. This is because if the latter holds then according to the above long exact sequence, x^{n−1}z is in the image of transfer. If x^{n−1}z = Tr_{N,G}(y) then d_{n}[1 ⊗ y] = [x^{n} ⊗ z] ≠ 0 in E_{n}^{n,∗}.

3 The Example

According to the analysis of the last section, in order to produce an example of a split extension

1 → N → G → Z/2 → 1

with d_{n} ≠ 0 in the corresponding spectral sequence, it suffices to produce a group G with elements x ∈ H^{1}(G, F_{2}) and z ∈ H^{∗}(G, k) with x non-nilpotent, and x^{n}z = 0 but x^{n−1}z ≠ 0. We take for G a nilpotent group of class two of the form

1 → (Z/2)^{n} → G → (Z/2)^{n+1} → 1.

The center Z(G) is an elementary abelian group of order 2^{n} generated by commuting involutions h_{1}, …, h_{n}. The group G is generated by these (redundant) generators together with generators g_{0}, …, g_{n} commuting with h_{1}, …, h_{n}, and satisfying g_{0}^{2} = 1, g_{1}^{2} = 1, g_{j}^{2} = h_{j} (j ≥ 2), [g_{0}, g_{1}] = h_{1}, [g_{j}, g_{j+1}] = h_{j} (j ≥ 1), and the remaining commutators [g_{i}, g_{j}] are equal to 1.

We have H^{∗}(G/Z(G), k) = k[x_{0}, …, x_{n}], H^{∗}(Z(G), k) = k[y_{1}, …, y_{n}], where the x’s and y’s are the dual bases to the g’s and h’s, in H^{1}(G/Z(G), k) ≅ Hom(G/Z(G), k) and H^{1}(Z(G), k) ≅ Hom(Z(G), k), respectively.

In the spectral sequence

H^{∗}(G/Z(G), H^{∗}(Z(G), k)) ⇒ H^{∗}(G, k)

we have

E_{2}^{∗∗} = k[x_{0}, …, x_{n}] ⊗ k[y_{1}, …, y_{n}]
d_{2}(y_{1}) = x_{0}x_{1} + x_{1}x_{2}
d_{2}(y_{j}) = x_{j}^{2} + x_{j}x_{j+1}  (2 ≤ j ≤ n − 1)
d_{2}(y_{n}) = x_{n}^{2}.

It is easy to check using Macaulay’s criterion (Zariski and Samuel [5], Appendix 6, Theorem 2) that d_{2}(y_{1}), …, d_{2}(y_{n}) form a regular sequence in k[x_{0}, …, x_{n}], and so

E_{3}^{∗∗} = k[x_{0}, …, x_{n}]/(d_{2}(y_{1}), …, d_{2}(y_{n})) ⊗ k[y_{1}^{2}, …, y_{n}^{2}].

Now for each j (1 ≤ j ≤ n), d_{3}(y_{j}^{2}) = Sq^{1}d_{2}(y_{j}) is in the ideal generated by d_{2}(y_{j}), and so d_{3} = 0. It follows that all further differentials are zero, and there are no ungrading problems, and so H^{∗}(G, k) ≅ E_{∞} ≅ E_{3}.

The Poincaré series of the ring k[x_{0}, …, x_{n}]/(d_{2}(y_{1}), …, d_{2}(y_{n})) is (1−t^{2})^{n}/(1−t)^{n+1} = (1 + t)^{n}/(1 − t). The relations allow every element to be expressed as a linear combination of monomials x_{0}^{α_{0}}x_{1}^{α_{1}} … x_{n}^{α_{n}} with either α_{0} = 0 or α_{1} = 0, and each α_{j} = 0 or 1 for 2 ≤ j ≤ n. The Poincaré series then forces these elements to be linearly independent.

So

x_{0}^{n}x_{1} = x_{1}x_{2}^{n} = x_{1}x_{2}x_{3}^{n−1} = ⋯ = x_{1}x_{2} … x_{n}^{2} = 0.

while

x_{0}^{n−1}x_{1} = x_{1}x_{2}^{n−1} = x_{1}x_{2}x_{3}^{n−2} = ⋯ = x_{1}x_{2} … x_{n} ≠ 0

is an element which is linearly independent of x_{0}^{n} and hence not divisible by x_{0}^{n}. So we may take x to be x_{0} and z to be x_{1}. In other words, N is the normal subgroup of index two generated by g_{1}, …, g_{n}, h_{1}, …, h_{n}, and x_{0}^{n} ⊗ x_{1} is in the image of d_{n} in the spectral sequence

H^{∗}(G/N, H^{∗}(N, k)) ⇒ H^{∗}(G, k).

References

[1] G. Avrunin and J. F. Carlson. Nilpotency degree of cohomology rings in characteristic two. To appear in Proc. Amer. Math. Soc.

[2] D. J. Benson. Representations and cohomology II: Cohomology of groups and modules. Cambridge studies in advanced mathematics 31, C.U.P., 1991.

[3] M. Feshbach. The image of H^{∗}(BG; Z) in H^{∗}(BT; Z) for G a compact Lie group with maximal torus T. Topology 20 (1981), 93–95.

[4] D. Quillen. The spectrum of an equivariant cohomology ring, I, II. Ann. of Math. 94 (1971), 549–602.

[5] O. Zariski and P. Samuel. Commutative Algebra, Volume II. Graduate Texts in Mathematics 29, Springer-Verlag, Berlin/New York, 1975.