Projectivity and cohomological triviality

Let k be a commutative ring and G a group. A representation A of G over k whose k[G]-module is projective has vanishing Tate cohomology in every degree on every finite subgroup of G (Serre, Local Fields, IX §5; Brown, Cohomology of Groups, VI §8). More generally, so does a representation of projective dimension at most one over k[G]. This is the easy half of the theorem of Nakayama and Rim, which for k = ℤ characterizes the cohomologically trivial G-modules of a finite group G as those of projective dimension at most one over ℤ[G]. Both statements survive tensoring, with the diagonal action, by an arbitrary representation M: M ⊗ A is cohomologically trivial for projective A, and for A with a projective resolution 0 → P₁ → P₀ → A → 0 of length one as soon as M ⊗ P₁ → M ⊗ P₀ stays injective. This is the form in which cohomological triviality enters the Tate–Nakayama generalization of Tate's theorem.

The projection Ind_⊥^G A → A from the representation induced from the trivial subgroup is an epimorphism, so a projective A is a retract of Ind_⊥^G A, and M ⊗ A is a retract of M ⊗ Ind_⊥^G A. The Tate cohomology of every finite subgroup with coefficients in M ⊗ Ind_⊥^G A vanishes (TauCeti.TateCohomology.isZero_tensor_indBot, after restricting the induced module to the subgroup), hence so does that of its retract M ⊗ A; for M the unit representation this is the statement for A itself. If 0 → P₁ → P₀ → A → 0 is exact with P₀ and P₁ projective, the Tate cohomology of A sits in the long exact sequence between that of P₀ and that of P₁, both of which vanish; the same holds after tensoring with M whenever the tensored sequence is still short exact.

Conversely, let k be an integral domain of characteristic zero in which every prime number is a unit or generates a maximal ideal, for instance ℤ, ℤ_[p], ℤ_(p) or a field of characteristic zero, and let G be finite. A cohomologically trivial representation A of G over k whose underlying k-module is projective (for instance free, of any rank) is a projective k[G]-module: this is the lattice form of the theorem of Nakayama and Rim.

It suffices that the identity of A is a norm ∑ g, A.ρ g ∘ φ ∘ A.ρ g⁻¹ (Rep.moduleProjective_of_id_mem_range_norm_linHom), and this may be checked on a Sylow p-subgroup P for each prime p (Representation.id_mem_range_norm_linHom_of_forall_prime). If p is a unit in k, the identity is |P|⁻¹ times the norm of the identity. Otherwise F = k/pk is a field of characteristic p. The vanishing of H_Tate⁰(P, A) and H_Tate⁻¹(P, A) gives the vanishing of H_Tate⁻¹(P, A/pA) (Rep.ker_norm_baseChange_le_coinvariantsKer), so the identity of A/pA is a norm (Representation.id_mem_range_norm_linHom_of_ker_norm_le), and so is the identity of A (Representation.id_mem_range_norm_linHom_of_baseChange). Only primes dividing |G|, and only the degrees 0 and -1 on one Sylow subgroup for each of them, are used (Rep.projective_of_isZero_res_sylow).

Main statements

• Rep.isZero_res_tensor_of_projective: if P.ρ.asModule is a projective k[G]-module, then H-hat^n(S, M ⊗ P) = 0 for every representation M, every finite subgroup S of G and every n : ℤ.
• Rep.isZero_res_of_projective: if A.ρ.asModule is a projective k[G]-module, then H-hat^n(S, A) = 0 for every finite subgroup S of G and every n : ℤ.
• Rep.isZero_res_ofModuleMonoidAlgebra_obj: the same for the representation attached to a projective k[G]-module.
• Rep.ker_norm_baseChange_le_coinvariantsKer: if H_Tate⁰(G, A) = H_Tate⁻¹(G, A) = 0 and p is regular on A, then H_Tate⁻¹(G, (k/pk) ⊗ A) = 0, in the form ker N ≤ I_G ((k/pk) ⊗ A).
• Rep.projective_of_isZero_res_sylow: a representation of a finite group on a projective k-module is projective over k[G] if, for each prime p dividing |G|, either p is a unit in k, or pk is maximal and H_Tate⁰ and H_Tate⁻¹ vanish on a Sylow p-subgroup.
• Rep.projective_of_isZero_res: over k as above, a cohomologically trivial representation of a finite group whose underlying k-module is projective is projective over k[G].
• Rep.isZero_res_of_exact: the same vanishing when A.ρ.asModule has a projective resolution 0 → P₁ → P₀ → A → 0 of length one.
• Rep.isZero_res_tensor_of_shortExact: H-hat^n(S, M ⊗ A) = 0 when A has a projective resolution 0 → P₁ → P₀ → A → 0 of length one with M ⊗ P₁ → M ⊗ P₀ injective.

References

• J.-P. Serre, Local Fields, Chapter IX, §§3–5.
• K. S. Brown, Cohomology of Groups, Chapter VI, §8.
• D. S. Rim, Modules over finite groups, Ann. of Math. 69 (1959).

theorem Rep.isZero_res_tensor_of_projective {k G : Type u} [CommRing k] [Group G] (M P : Rep.{u, u, u} k G) [Module.Projective (MonoidAlgebra k G) P.ρ.asModule] (S : Subgroup G) [Fintype ↥S] (n : ℤ) : CategoryTheory.Limits.IsZero (tateCohomology (res S.subtype (CategoryTheory.MonoidalCategoryStruct.tensorObj M P)) n)

Tensoring with a projective module gives a cohomologically trivial module. If the k[G]-module of a representation P is projective, then for every representation M the Tate cohomology of every finite subgroup S of G with coefficients in M ⊗ P vanishes in every degree.

theorem Rep.isZero_res_of_projective {k G : Type u} [CommRing k] [Group G] (A : Rep.{u, u, u} k G) [Module.Projective (MonoidAlgebra k G) A.ρ.asModule] (S : Subgroup G) [Fintype ↥S] (n : ℤ) : CategoryTheory.Limits.IsZero (tateCohomology (res S.subtype A) n)

Projective modules are cohomologically trivial. If the k[G]-module of a representation A is projective, then the Tate cohomology of every finite subgroup S of G with coefficients in A vanishes in every degree.

theorem Rep.isZero_res_ofModuleMonoidAlgebra_obj {k G : Type u} [CommRing k] [Group G] (M : ModuleCat (MonoidAlgebra k G)) [Module.Projective (MonoidAlgebra k G) ↑M] (S : Subgroup G) [Fintype ↥S] (n : ℤ) : CategoryTheory.Limits.IsZero (tateCohomology (res S.subtype (ofModuleMonoidAlgebra.obj M)) n)

The representation attached to a projective k[G]-module is cohomologically trivial: its Tate cohomology on every finite subgroup S of G vanishes in every degree.

theorem Rep.ker_norm_baseChange_le_coinvariantsKer {k G : Type u} [CommRing k] [Group G] [Fintype G] (A : Rep.{u, u, u} k G) (p : k) (hp : ∀ (v : ↑A), p • v = 0 → v = 0) (h0 : CategoryTheory.Limits.IsZero (tateCohomology A 0)) (h1 : CategoryTheory.Limits.IsZero (tateCohomology A (-1))) : LinearMap.ker (Representation.baseChange (k ⧸ Ideal.span {p}) A.ρ).norm ≤ Representation.Coinvariants.ker (Representation.baseChange (k ⧸ Ideal.span {p}) A.ρ)

H_Tate⁻¹ modulo p. If H_Tate⁰(G, A) and H_Tate⁻¹(G, A) vanish and multiplication by p is injective on A, then every vector of (k/pk) ⊗ A of norm zero lies in the augmentation submodule; that is, H_Tate⁻¹(G, (k/pk) ⊗ A) = 0.

theorem Rep.projective_of_isZero_res_sylow {k G : Type u} [CommRing k] [Group G] [Finite G] (A : Rep.{u, u, u} k G) [Module.Projective k ↑A] (h : ∀ (p : ℕ), Nat.Prime p → p ∣ Nat.card G → ∃ (P : Sylow p G) (x : Fintype ↥↑P), IsUnit ↑p ∨ (Ideal.span {↑p}).IsMaximal ∧ (∀ (v : ↑A), ↑p • v = 0 → v = 0) ∧ CategoryTheory.Limits.IsZero (tateCohomology (res (↑P).subtype A) 0) ∧ CategoryTheory.Limits.IsZero (tateCohomology (res (↑P).subtype A) (-1))) : Module.Projective (MonoidAlgebra k G) A.ρ.asModule

Nakayama–Rim, Sylow form (Serre, Local Fields, IX §§3–5; Rim, Ann. of Math. 69 (1959)). Let k be a commutative ring, G a finite group, and A a representation of G on a projective k-module. Suppose that for each prime p dividing |G| some Sylow p-subgroup P satisfies: p is a unit in k, or pk is a maximal ideal, multiplication by p is injective on A, and H_Tate⁰(P, A) = H_Tate⁻¹(P, A) = 0. Then A is projective over k[G].

theorem Rep.projective_of_isZero_res {k G : Type u} [CommRing k] [Group G] [Finite G] [IsDomain k] [CharZero k] (hk : ∀ (p : ℕ), Nat.Prime p → IsUnit ↑p ∨ (Ideal.span {↑p}).IsMaximal) (A : Rep.{u, u, u} k G) [Module.Projective k ↑A] (hA : ∀ (S : Subgroup G) [inst : Fintype ↥S] (n : ℤ), CategoryTheory.Limits.IsZero (tateCohomology (res S.subtype A) n)) : Module.Projective (MonoidAlgebra k G) A.ρ.asModule

Nakayama–Rim, lattice form (Serre, Local Fields, IX §§3–5; Rim, Ann. of Math. 69 (1959)). Over an integral domain of characteristic zero in which every prime number is a unit or generates a maximal ideal (for instance ℤ, ℤ_[p] or ℤ_(p)), a cohomologically trivial representation of a finite group whose underlying k-module is projective — for instance free, of any rank — is projective over k[G].

theorem Rep.isZero_res_of_exact {k G : Type u} [CommRing k] [Group G] (A : Rep.{u, u, u} k G) {P₀ P₁ : Type u} [AddCommGroup P₀] [Module (MonoidAlgebra k G) P₀] [Module.Projective (MonoidAlgebra k G) P₀] [AddCommGroup P₁] [Module (MonoidAlgebra k G) P₁] [Module.Projective (MonoidAlgebra k G) P₁] {d : P₁ →ₗ[MonoidAlgebra k G] P₀} {q : P₀ →ₗ[MonoidAlgebra k G] A.ρ.asModule} (hd : Function.Injective ⇑d) (hdq : Function.Exact ⇑d ⇑q) (hq : Function.Surjective ⇑q) (S : Subgroup G) [Fintype ↥S] (n : ℤ) : CategoryTheory.Limits.IsZero (tateCohomology (res S.subtype A) n)

Projective dimension at most one implies cohomological triviality. Let 0 → P₁ → P₀ → A → 0 be an exact sequence of k[G]-modules with P₀ and P₁ projective. Then the Tate cohomology of every finite subgroup S of G with coefficients in A vanishes in every degree.

theorem Rep.isZero_res_tensor_of_shortExact {k G : Type u} [CommRing k] [Group G] (M : Rep.{u, u, u} k G) {T : CategoryTheory.ShortComplex (Rep.{u, u, u} k G)} (hT : T.ShortExact) [Module.Projective (MonoidAlgebra k G) T.X₁.ρ.asModule] [Module.Projective (MonoidAlgebra k G) T.X₂.ρ.asModule] (hf : Function.Injective ⇑(LinearMap.lTensor (↑M) (Hom.hom T.f).toLinearMap)) (S : Subgroup G) [Fintype ↥S] (n : ℤ) : CategoryTheory.Limits.IsZero (tateCohomology (res S.subtype (CategoryTheory.MonoidalCategoryStruct.tensorObj M T.X₃)) n)

Tensoring a resolution of length one. Let 0 → P₁ → P₀ → A → 0 be a short exact sequence of representations whose first two terms are projective k[G]-modules, and suppose that M ⊗ P₁ → M ⊗ P₀ is injective. Then for every finite subgroup S of G the Tate cohomology of S with coefficients in M ⊗ A vanishes in every degree.