Plain-language statement
Given a 2-cocycle σ, the image of σ in the splitting module of σ is equal to the coboundary of τ σ.
Exact Lean statement
lemma τ_property (g h : G) :
(split σ).ρ g (τ σ h) - τ σ (g * h) + τ σ g = ι σ (cocycle σ (g,h))Formal artifact
Lean source
lemma τ_property (g h : G) : (split σ).ρ g (τ σ h) - τ σ (g * h) + τ σ g = ι σ (cocycle σ (g,h)) := by classical rw [τ, apply, τ, τ, ι_apply] ext · simp only [equalizer_as_kernel, map_mul, Module.End.mul_apply, add_fst, sub_fst] apply (Rep.mono_iff_injective _).mp (inferInstance : (Mono (aug.ι R G))) simp only [equalizer_as_kernel, map_add, map_sub, aug.ofSubOfOne_spec R G, map_zero] rw [Rep.hom_comm_apply, Rep.aug.ofSubOfOne_spec] simp · classical simp only [equalizer_as_kernel, Rep.aug.ofSubOfOne_spec R G, MonoidAlgebra.coeff_sub, MonoidAlgebra.coeff_single, Finsupp.coe_sub, Pi.sub_apply, Finsupp.single_apply, sub_smul, ite_smul, one_smul, zero_smul, Finset.sum_sub_distrib, Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, map_mul, Module.End.mul_apply, add_snd, sub_snd, add_sub_cancel_left] have : (cocycle σ) (g, 1) = (M.ρ g) ((cocycle σ) (1, 1)) := by simpa [add_comm] using (mem_cocycles₂_iff (cocycle σ)).mp (cocycle σ).2 g 1 1 simp [this]- Project
- Class Field Theory
- License
- Apache-2.0
- Commit
- f18cd7fd1575
- Source
- ClassFieldTheory/Cohomology/SplittingModule.lean:199-216
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