The Tate–Nakayama theorem #
Let G be a finite group, C a representation of G over ℤ and u ∈ H²(G, C). Suppose that
for every subgroup S of G of prime-power order, H¹(S, C) = 0, the restriction of u
generates H²(S, C), and H²(S, C) has |S| elements. Then for every representation M with
Tor₁^ℤ(M, C) = 0, cup product with u is a bijection
H^r(G, M) → H^{r+2}(G, M ⊗ C)
in every integer degree r
(TauCeti.TateCohomology.cup_bijective_of_forall_isPGroup_of_lTensor_injective). This is
Nakayama's generalization (Cohomology of class field theory and tensor product modules I,
Ann. of Math. 65 (1957)) of Tate's theorem, the case M = ℤ
(TauCeti.TateCohomology.cup_bijective_of_forall_isPGroup). As in
Rep.isZero_res_tensor_of_isZero_res, the vanishing of Tor₁^ℤ(M, C) is stated without a Tor
functor: M ⊗ X → M ⊗ Y is injective for every short exact sequence 0 → X → Y → C → 0 of
abelian groups. It holds for instance whenever M or C is torsion-free. Throughout this
file, H^n(G, M) denotes Tate cohomology in the integer degree n.
Proof #
The class u sits in two short exact sequences, both split over ℤ: the splitting-module sequence
0 → C → C(u) → I_G → 0 (Rep.splittingModule) and the augmentation sequence
0 → I_G → ℤ[G] → ℤ → 0. The splitting module is cohomologically trivial under Tate's hypotheses
(TauCeti.TateCohomology.isZero_res_splittingModule), and so is M ⊗ C(u): as an abelian group
C(u) is I_G × C, so Tor₁^ℤ(M, C(u)) = Tor₁^ℤ(M, C) = 0, and the theorem of Nakayama and Rim
applies (TauCeti.TateCohomology.isZero_res_tensor_splittingModule). Since M ⊗ ℤ[G] is induced,
both tensored sequences have bijective connecting maps δ. As u dies in C(u), it is
δ (δ z) for some z ∈ H⁰(G, ℤ) = ℤ/|G|, and the rule x ∪ δ y = (-1)^r δ (x ∪ y) for ℤ-split
sequences (TauCeti.TateCohomology.cup_δ_of_leftInverse), applied twice, turns cup product with
u into x ↦ δ (δ (x ∪ z)). Finally z is the class of an integer prime to |G|: a prime
p ∣ |G| dividing it would make u restrict to zero on a subgroup of order p, whose H² it
generates and which has p elements. Such an integer is invertible in H⁰(G, ℤ) = ℤ/|G|, so cup
product with z is bijective, as cup product with the class of 1 is.
Main statements #
TauCeti.TateCohomology.isZero_res_tensor_splittingModule: under Tate's hypotheses andTor₁(M, C) = 0, the tensor productM ⊗ C(u)with the splitting module is cohomologically trivial.TauCeti.TateCohomology.cup_bijective_of_forall_isPGroup_of_lTensor_injective: the Tate–Nakayama theorem.
References #
- T. Nakayama, Cohomology of class field theory and tensor product modules I, Ann. of Math. 65 (1957).
- J.-P. Serre, Local Fields, Chapter IX, §8.
- K. S. Brown, Cohomology of Groups, Chapter VI, §8.
Tensoring the splitting module. Let k be a principal ideal domain of characteristic zero
in which every prime number is a unit or generates a maximal ideal, C a representation of a
finite group G over k and u ∈ H²(G, C). Suppose that for every subgroup S of G of
prime-power order, H¹(S, C) = 0, the restriction of u generates H²(S, C), and H²(S, C) has
the finite order of k ⧸ |S|k. If Tor₁^k(M, C) = 0, in the form that M ⊗ X → M ⊗ Y is
injective for every short exact sequence 0 → X → Y → C → 0 of k-modules, then
M ⊗ C(u) is cohomologically trivial.
The Tate–Nakayama theorem (Nakayama, Ann. of Math. 65 (1957); Serre, Local Fields, IX §8).
Let G be a finite group, C a representation of G over ℤ and u ∈ H²(G, C).
Suppose that for every subgroup S of G of prime-power order
H¹(S, C) = 0,- the restriction of
ugeneratesH²(S, C), and H²(S, C)has|S|elements.
Then for every representation M with Tor₁^ℤ(M, C) = 0, in the form that M ⊗ X → M ⊗ Y is
injective for every short exact sequence 0 → X → Y → C → 0 of abelian groups, cup product with
u is a bijection H^r(G, M) → H^{r+2}(G, M ⊗ C) in every integer degree r.