openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Complete /CompleteResolutionAllProjectives.lean
| import OAI.Algebra.AuslanderReiten.Complete.CompleteResolution | |
| import OAI.Algebra.AuslanderReiten.Basic | |
| import Mathlib.RingTheory.Finiteness.Finsupp | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit | |
| universe u v w x y | |
| variable {R : Type u} [Ring R] | |
| variable {K : Type v} [AddCommGroup K] [Module R K] | |
| variable {L : Type w} [AddCommGroup L] [Module R L] | |
| variable {M : Type x} [AddCommGroup M] [Module R M] | |
| variable {Q : Type y} [AddCommGroup Q] [Module R Q] | |
| theorem HomExactAt.arbitrary_projective [Module.Finite R L] [Module.Projective R Q] | |
| {e : K →ₗ[R] L} {d : L →ₗ[R] M} (h : HomExactAt e d) | |
| (f : L →ₗ[R] Q) (hf : f.comp e = 0) : | |
| ∃ g : M →ₗ[R] Q, g.comp d = f := by | |
| classical | |
| obtain ⟨s,hs⟩ := Module.projective_def'.mp (inferInstance : Module.Projective R Q) | |
| obtain ⟨c,hc⟩ := (LinearMap.finsuppLinearMap_bijective_of_moduleFinite | |
| R L R Q ℤ).surjective (s.comp f) | |
| have he (q : Q) : (c q).comp e = 0 := by | |
| ext z | |
| have hh := congrArg (fun F : L →ₗ[R] (Q →₀ R) => F (e z) q) hc | |
| change (c q) (e z) = (s (f (e z))) q at hh | |
| rw [show f (e z) = 0 from DFunLike.congr_fun hf z, map_zero] at hh | |
| exact hh | |
| choose g hg using fun q => h (c q) (he q) | |
| let G : M →ₗ[R] (Q →₀ R) := ∑ q ∈ c.support, (Finsupp.lsingle q).comp (g q) | |
| have hG : G.comp d = s.comp f := by | |
| ext z q | |
| have hh := congrArg (fun F : L →ₗ[R] (Q →₀ R) => F z q) hc | |
| change (c q) z = (s (f z)) q at hh | |
| change G (d z) q = (s (f z)) q | |
| rw [← hh] | |
| change (∑ j ∈ c.support, (Finsupp.lsingle j).comp (g j)) (d z) q = (c q) z | |
| simp only [LinearMap.sum_apply, LinearMap.comp_apply, Finsupp.finsetSum_apply] | |
| have hg' (j : Q) : g j (d z) = (c j) z := DFunLike.congr_fun (hg j) z | |
| simp only [hg', Finsupp.lsingle_apply] | |
| rw [Finset.sum_eq_single q] | |
| · simp | |
| · intro b _ hb | |
| simp [Ne.symm hb] | |
| · intro hq | |
| have hcz : c q = 0 := Finsupp.notMem_support_iff.mp hq | |
| simp [hcz] | |
| refine ⟨(Finsupp.linearCombination R id).comp G, ?_⟩ | |
| rw [LinearMap.comp_assoc, hG, ← LinearMap.comp_assoc, hs, LinearMap.id_comp] | |
| def GorensteinProjectiveAllTargets (R : Type u) [Ring R] | |
| (M : Type v) [AddCommGroup M] [Module R M] : Prop := | |
| ∃ P : ChainComplex (ModuleCat.{max u v} R) ℤ, | |
| (∀ n : ℤ, Module.Finite R (P.X n) ∧ Module.Projective R (P.X n)) ∧ | |
| (∀ n : ℤ, LinearMap.range (P.d (n+1) n).hom = LinearMap.ker (P.d n (n-1)).hom) ∧ | |
| (∀ (Q : Type y) [AddCommGroup Q] [Module R Q] [Module.Projective R Q] | |
| (n : ℤ) (f : P.X n →ₗ[R] Q), f.comp (P.d (n+1) n).hom = 0 → | |
| ∃ g : P.X (n-1) →ₗ[R] Q, f = g.comp (P.d n (n-1)).hom) ∧ | |
| Nonempty (M ≃ₗ[R] (P.X 0 ⧸ LinearMap.range (P.d 1 0).hom)) | |
| theorem GorensteinProjective.all_projective_targets | |
| {M : Type v} [AddCommGroup M] [Module R M] | |
| (h : GorensteinProjective R M) : GorensteinProjectiveAllTargets.{u,v,y} R M := by | |
| obtain ⟨P,hP,he,hd,hi⟩ := h | |
| refine ⟨P,hP,he,?_,hi⟩ | |
| intro Q _ _ _ n f hf | |
| let := (hP n).1 | |
| have hn : HomExactAt (P.d (n+1) n).hom (P.d n (n-1)).hom := by | |
| intro f hf | |
| obtain ⟨g,hg⟩ := hd n f hf | |
| exact ⟨g,hg.symm⟩ | |
| obtain ⟨g,hg⟩ := hn.arbitrary_projective f hf | |
| exact ⟨g,hg.symm⟩ | |
| theorem gorensteinProjective_iff_all_projective_targets | |
| {M : Type v} [AddCommGroup M] [Module R M] : | |
| GorensteinProjective R M ↔ GorensteinProjectiveAllTargets.{u,v,u} R M := by | |
| refine ⟨GorensteinProjective.all_projective_targets, ?_⟩ | |
| rintro ⟨P,hP,he,hd,hi⟩ | |
| exact ⟨P,hP,he,hd R,hi⟩ | |
| end ArExplicit | |
| end | |
| end OAI | |