Is Zero of Even odd
Rep.isZero_ofEven_odd
Plain-language statement
Let M be a representation of a finite cyclic group G. Suppose there are even and positive integers e and o with e even and o odd, such that Hᵉ(G,M) and Hᵒ(G,M) are both zero. Then Hⁿ(G,M) is zero for all n > 0.
Exact Lean statement
lemma isZero_ofEven_odd {e o : ℕ} [NeZero e] (he : Even e) (ho : Odd o)
(hₑ : IsZero (groupCohomology M e)) (hₒ : IsZero (groupCohomology M o)) (n : ℕ) [NeZero n] :
IsZero (groupCohomology M n)Formal artifact
Lean source
lemma isZero_ofEven_odd {e o : ℕ} [NeZero e] (he : Even e) (ho : Odd o) (hₑ : IsZero (groupCohomology M e)) (hₒ : IsZero (groupCohomology M o)) (n : ℕ) [NeZero n] : IsZero (groupCohomology M n) := by cases nonempty_fintype G obtain hn | hn := n.even_or_odd · refine .of_iso hₑ <| (periodicCohomology n e ?_).app M grind [Nat.modEq_iff_dvd] · have : NeZero o := ⟨ho.pos.ne'⟩ refine .of_iso hₒ <| (periodicCohomology n o ?_).app M grind [Nat.modEq_iff_dvd]- Project
- Class Field Theory
- License
- Apache-2.0
- Commit
- f18cd7fd1575
- Source
- ClassFieldTheory/Cohomology/FiniteCyclic/UpDown.lean:416-425
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