Infl δ naturality
groupCohomology.infl_δ_naturality
Plain-language statement
Assume that we have a short exact sequence 0 → A → B → C → 0 in Rep R G and that the sequence of H- invariants is also a short exact in Rep R (G ⧸ H) : 0 → Aᴴ → Bᴴ → Cᴴ → 0. Then we have a commuting square Hⁿ(G ⧸ H, Cᴴ) ⟶ H^{n+1}(G ⧸ H, Aᴴ) | | ↓ ↓ Hⁿ(G , C) ⟶ H^{n+1}(G,A) where the horizontal maps are connecting homomorphisms an...
Exact Lean statement
lemma infl_δ_naturality {S : ShortComplex (Rep R G)} (hS : S.ShortExact)
(hS' : (S.map (quotientToInvariantsFunctor' surj)).ShortExact) (i j : ℕ) (hij : i + 1 = j) :
δ hS' i j hij ≫ (infl surj j).app _ = (infl surj i).app _ ≫ δ hS i j hijFormal artifact
Lean source
lemma infl_δ_naturality {S : ShortComplex (Rep R G)} (hS : S.ShortExact) (hS' : (S.map (quotientToInvariantsFunctor' surj)).ShortExact) (i j : ℕ) (hij : i + 1 = j) : δ hS' i j hij ≫ (infl surj j).app _ = (infl surj i).app _ ≫ δ hS i j hij := by let C := S.map (cochainsFunctor R G) let S' := S.map (quotientToInvariantsFunctor' surj) let C' := S'.map (cochainsFunctor R Q) let φ : C' ⟶ C := { τ₁ := (cochain_infl surj).app S.X₁ τ₂ := (cochain_infl surj).app S.X₂ τ₃ := (cochain_infl surj).app S.X₃ comm₁₂ := ((cochain_infl surj).naturality S.f).symm comm₂₃ := ((cochain_infl surj).naturality S.g).symm } have ses₁ : C.ShortExact := map_cochainsFunctor_shortExact hS have ses₂ : C'.ShortExact := map_cochainsFunctor_shortExact hS' exact HomologySequence.δ_naturality φ ses₂ ses₁ i j hij- Project
- Class Field Theory
- License
- Apache-2.0
- Commit
- f18cd7fd1575
- Source
- ClassFieldTheory/Cohomology/Functors/Inflation.lean:131-147
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Continuous algebra Map of density
continuous_algebraMap_of_density
Plain-language statement
Maddy's Lemma : Density implies continuity.
Source project: Class Field Theory
Person-level attribution pending.
Exists valuation algebra Map eq valuation pow
exists_valuation_algebraMap_eq_valuation_pow
Plain-language statement
Andrew's Lemma : Density for algebraic extensions.
Source project: Class Field Theory
Person-level attribution pending.
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.
Source project: Class Field Theory
Person-level attribution pending.