Trivial Tate Cohomology of cases
Rep.TrivialTateCohomology.of_cases
Plain-language statement
To check that a finite group has trivial Tate cohomology, it's enough to show it has trivial cohomology and trivial homology, and that the 0-th and -1st Tate cohomology groups are trivial.
Exact Lean statement
lemma TrivialTateCohomology.of_cases [Finite G] {M : Rep R G}
[M.TrivialCohomology] [M.TrivialHomology]
(h : ∀ (H : Subgroup G),
letI : Fintype H := Fintype.ofFinite _
IsZero (tateCohomology (M ↓ H.subtype : Rep R H) 0) ∧
IsZero (tateCohomology (M ↓ H.subtype : Rep R H) (-1))) :
TrivialTateCohomology M where
isZero H nFormal artifact
Lean source
lemma TrivialTateCohomology.of_cases [Finite G] {M : Rep R G} [M.TrivialCohomology] [M.TrivialHomology] (h : ∀ (H : Subgroup G), letI : Fintype H := Fintype.ofFinite _ IsZero (tateCohomology (M ↓ H.subtype : Rep R H) 0) ∧ IsZero (tateCohomology (M ↓ H.subtype : Rep R H) (-1))) : TrivialTateCohomology M where isZero H n := by match n with | .ofNat (n + 1) => exact isZero_of_trivialCohomology.of_iso <| letI : Fintype H := Fintype.ofFinite _ (TateCohomology.isoGroupCohomology (n + 1)).app (M ↓ H.subtype) | .negSucc (n + 1) => letI := TrivialHomology.res M (H := H) H.subtype_injective exact isZero_of_trivialHomology.of_iso <| letI : Fintype H := Fintype.ofFinite _ (TateCohomology.isoGroupHomology _ (n + 1) rfl).app (M ↓ H.subtype) | 0 => aesop | .negSucc 0 => aesop- Project
- Class Field Theory
- License
- Apache-2.0
- Commit
- f18cd7fd1575
- Source
- ClassFieldTheory/Cohomology/TrivialCohomology.lean:152-173
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