Representations of p-groups with vanishing degree -1 Tate cohomology

Let G be a finite p-group and ρ a representation of G on a vector space V over a field of characteristic p. If every vector of norm zero lies in the augmentation submodule — that is, if the degree -1 Tate cohomology of G with coefficients in V vanishes — then the identity of V is a norm x ↦ ∑ g, ρ g (φ (ρ g⁻¹ x)) for the conjugation action of G on End(V) (Serre, Local Fields, IX §3, Theorem 4; Brown, Cohomology of Groups, VI 8.5). No finiteness is assumed on V. In the theorem of Nakayama and Rim, this provides the identity-norm condition in characteristic p, which Representation.id_mem_range_norm_linHom_of_baseChange then lifts.

Main statements

• Representation.id_mem_range_norm_linHom_of_ker_norm_le: for a finite p-group in characteristic p, if the kernel of the norm lies in the augmentation submodule, then the identity is a norm for the conjugation action on End(V).

References

• J.-P. Serre, Local Fields, Chapter IX, §3.
• K. S. Brown, Cohomology of Groups, Chapter VI, §8.

theorem Representation.id_mem_range_norm_linHom_of_ker_norm_le {F : Type u_1} {G : Type u_2} {V : Type u_3} [Field F] [Group G] [Fintype G] [AddCommGroup V] [Module F V] (p : ℕ) [Fact (Nat.Prime p)] [CharP F p] (hG : IsPGroup p G) (ρ : Representation F G V) (h : LinearMap.ker ρ.norm ≤ Coinvariants.ker ρ) : LinearMap.id ∈ LinearMap.range (ρ.linHom ρ).norm

A representation of a p-group in characteristic p with vanishing degree -1 Tate cohomology has the identity as a norm. If G is a finite p-group, F has characteristic p, and every vector of V of norm zero lies in the augmentation submodule, then the identity is the norm ∑ g, ρ g ∘ φ ∘ ρ g⁻¹ of an F-linear φ.