Exists of surjective
groupCohomology.exists_of_surjective
Plain-language statement
Given map f: M ⟶ N and q : ℕ, if H^{q+1}(M) ⟶ H^{q+1}(N) is surjective, then any z : Z^{q+1}(N) can be written as f(z') + d(y) for some z' : Z^{q+1}(M) and y : C^q(M). Note that d is spelled as toCocycles.
Exact Lean statement
theorem exists_of_surjective {k G : Type u} [CommRing k] [Group G] {M N : Rep k G}
{f : M ⟶ N} {q : ℕ} (h : Function.Surjective (map (MonoidHom.id G) f (q + 1)))
(z : cocycles N (q + 1)) :
∃ z' : cocycles M (q + 1), ∃ y : (inhomogeneousCochains N).X q,
cocyclesMap (.id G) f (q + 1) z' + toCocycles N q (q + 1) y = zFormal artifact
Lean source
theorem exists_of_surjective {k G : Type u} [CommRing k] [Group G] {M N : Rep k G} {f : M ⟶ N} {q : ℕ} (h : Function.Surjective (map (MonoidHom.id G) f (q + 1))) (z : cocycles N (q + 1)) : ∃ z' : cocycles M (q + 1), ∃ y : (inhomogeneousCochains N).X q, cocyclesMap (.id G) f (q + 1) z' + toCocycles N q (q + 1) y = z := by have hc₁ := (inhomogeneousCochains N).homologyIsCokernel q (q + 1) (by simp) have hc₂ := ModuleCat.cokernelIsColimit ((inhomogeneousCochains N).toCycles q (q + 1)) obtain ⟨z', hz'⟩ := h (π N (q + 1) z) induction z' using groupCohomology_induction_on with | h z' => rw [π_map_apply] at hz' obtain ⟨y, hy⟩ := exists_of_π_eq_π hz' exact ⟨z', y, hy⟩- Project
- Class Field Theory
- License
- Apache-2.0
- Commit
- f18cd7fd1575
- Source
- ClassFieldTheory/Cohomology/SerreApproximation.lean:269-280
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