| import OAI.Algebra.AuslanderReiten.Duality.DualResolution | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian | |
| namespace ArExplicit.Resolution | |
| open ArExplicit.Letter ArExplicit.BaseAlgebra | |
| variable {k : Type*} [Field k] [CharP k 2] (q : k) | |
| local notation "C" => BaseAlgebra k q | |
| theorem dualHom_exact_one (g : E q →ₗ[C] C) (hg : g.comp (dEll q 1) = 0) : | |
| ∃ f : F q →ₗ[C] C, f.comp (dU q) = g := by | |
| have hc : leftEll q 1 (evaluateE q g) = 0 := by | |
| rw [← evaluateE_precompose, hg] | |
| rfl | |
| obtain ⟨a, ha⟩ := (leftU_exact q (evaluateE q g)).mp hc | |
| refine ⟨fromRightF q a, ?_⟩ | |
| apply (homEEquiv q).injective | |
| change evaluateE q ((fromRightF q a).comp (dU q)) = evaluateE q g | |
| rw [evaluate_precompose_dU, evaluateF_fromRightF, ha] | |
| theorem dualHom_exact_tail (r : k) (hq : q ≠ 0) (hr : 1 + r ≠ 0) | |
| (hqr : 1 + q * r ≠ 0) (g : E q →ₗ[C] C) | |
| (hg : g.comp (dEll q (q * r)) = 0) : | |
| ∃ f : E q →ₗ[C] C, f.comp (dEll q r) = g := by | |
| have hc : leftEll q (q * r) (evaluateE q g) = 0 := by | |
| rw [← evaluateE_precompose, hg] | |
| rfl | |
| obtain ⟨a, ha⟩ := (leftEll_exact q r hq hr hqr (evaluateE q g)).mp hc | |
| refine ⟨fromRightE q a, ?_⟩ | |
| apply (homEEquiv q).injective | |
| change evaluateE q ((fromRightE q a).comp (dEll q r)) = evaluateE q g | |
| rw [evaluateE_precompose, evaluateE_fromRightE, ha] | |
| omit [CharP k 2] in | |
| theorem hom_simple_regular_eq_zero (g : Simple q →ₗ[C] C) : g = 0 := by | |
| let f := g.comp (LinearMap.range (dU q)).mkQ | |
| have hf : f.comp (dU q) = 0 := by | |
| rw [LinearMap.comp_assoc, LinearMap.range_mkQ_comp, LinearMap.comp_zero] | |
| have hz : evaluateF q f = 0 := by | |
| apply leftU_injective q | |
| rw [← evaluate_precompose_dU, hf, map_zero] | |
| rfl | |
| have hf0 : f = 0 := (homFEquiv q).injective hz | |
| apply LinearMap.ext | |
| intro a | |
| refine Submodule.Quotient.induction_on _ a ?_ | |
| intro b | |
| exact LinearMap.congr_fun hf0 b | |
| omit [CharP k 2] in | |
| theorem regular_ext_zero : | |
| Subsingleton (Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) 0) := by | |
| refine ⟨fun a b => ?_⟩ | |
| apply Ext.homEquiv₀.injective | |
| apply ModuleCat.hom_ext | |
| exact (hom_simple_regular_eq_zero q (Ext.homEquiv₀ a).hom).trans | |
| (hom_simple_regular_eq_zero q (Ext.homEquiv₀ b).hom).symm | |
| theorem regular_ext_one (h : NoPositivePowerOne q) : | |
| Subsingleton (Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) 1) := by | |
| let P := projectiveResolution q h | |
| have hz : ∀ a : Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) 1, a = 0 := by | |
| intro a | |
| obtain ⟨g, hg, rfl⟩ := P.extMk_surjective a 2 rfl | |
| apply (P.extMk_eq_zero_iff g 2 rfl hg 0 rfl).mpr | |
| have hg' : g.hom.comp (dEll q 1) = 0 := by | |
| have hh := congrArg ModuleCat.Hom.hom hg | |
| change g.hom.comp (dEll q (q ^ 0)) = 0 at hh | |
| simpa only [pow_zero] using hh | |
| obtain ⟨f, hf⟩ := dualHom_exact_one q g.hom hg' | |
| refine ⟨ModuleCat.ofHom f, ?_⟩ | |
| apply ModuleCat.hom_ext | |
| exact hf | |
| exact ⟨fun a b => (hz a).trans (hz b).symm⟩ | |
| theorem regular_ext_tail (h : NoPositivePowerOne q) (hq : q ≠ 0) (n : ℕ) : | |
| Subsingleton (Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) (n + 3)) := by | |
| let P := projectiveResolution q h | |
| have hz : ∀ a : Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) (n + 3), a = 0 := by | |
| intro a | |
| obtain ⟨g, hg, rfl⟩ := P.extMk_surjective a (n + 4) rfl | |
| apply (P.extMk_eq_zero_iff g (n + 4) rfl hg (n + 2) rfl).mpr | |
| have hg' : g.hom.comp (dEll q (q * q ^ (n + 1))) = 0 := by | |
| have hh := congrArg ModuleCat.Hom.hom hg | |
| change (ChainComplex.of.d (objects q) (differential q) ((n + 3) + 1) (n + 3) ≫ g).hom = 0 at hh | |
| rw [ChainComplex.of_d] at hh | |
| change g.hom.comp (dEll q (q ^ (n + 2))) = 0 at hh | |
| simpa only [pow_succ'] using hh | |
| obtain ⟨f, hf⟩ := dualHom_exact_tail q (q ^ (n + 1)) hq | |
| (h _ (by omega)) (by simpa [← pow_succ'] using h (n + 2) (by omega)) g.hom hg' | |
| refine ⟨ModuleCat.ofHom f, ?_⟩ | |
| apply ModuleCat.hom_ext | |
| change (ChainComplex.of.d (objects q) (differential q) ((n + 2) + 1) (n + 2) ≫ ModuleCat.ofHom f).hom = g.hom | |
| rw [ChainComplex.of_d] | |
| change f.comp (dEll q (q ^ (n + 1))) = g.hom | |
| exact hf | |
| exact ⟨fun a b => (hz a).trans (hz b).symm⟩ | |
| theorem regular_ext_vanish (h : NoPositivePowerOne q) (hq : q ≠ 0) | |
| (n : ℕ) (hn : n ≠ 2) : | |
| Subsingleton (Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) n) := by | |
| match n with | |
| | 0 => exact regular_ext_zero q | |
| | 1 => exact regular_ext_one q h | |
| | 2 => exact (hn rfl).elim | |
| | n + 3 => exact regular_ext_tail q h hq n | |
| def exceptionalHom : E q →ₗ[C] C := fromRightE q (exceptionalCycle q) | |
| omit [CharP k 2] in | |
| theorem exceptionalHom_cocycle : (exceptionalHom q).comp (dEll q q) = 0 := by | |
| apply (homEEquiv q).injective | |
| change evaluateE q ((exceptionalHom q).comp (dEll q q)) = 0 | |
| rw [evaluateE_precompose] | |
| simpa only [exceptionalHom, evaluateE_fromRightE] using exceptionalCycle_closed q q | |
| theorem exceptionalHom_not_boundary (hq : q ≠ 0) (hq1 : 1 + q ≠ 0) : | |
| ¬ ∃ f : E q →ₗ[C] C, f.comp (dEll q 1) = exceptionalHom q := by | |
| rintro ⟨f, hf⟩ | |
| apply exceptionalCycle_not_boundary q hq hq1 | |
| refine ⟨evaluateE q f, ?_⟩ | |
| rw [← evaluateE_precompose, hf] | |
| exact evaluateE_fromRightE q (exceptionalCycle q) | |
| def regular_ext_two_class (h : NoPositivePowerOne q) : | |
| Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) 2 := | |
| (projectiveResolution q h).extMk (ModuleCat.ofHom (exceptionalHom q)) 3 rfl (by | |
| apply ModuleCat.hom_ext | |
| change (exceptionalHom q).comp (dEll q (q ^ 1)) = 0 | |
| simpa only [pow_one] using exceptionalHom_cocycle q) | |
| theorem regular_ext_two_class_ne_zero (h : NoPositivePowerOne q) (hq : q ≠ 0) : | |
| regular_ext_two_class q h ≠ 0 := by | |
| intro hz | |
| obtain ⟨f, hf⟩ := ((projectiveResolution q h).extMk_eq_zero_iff _ 3 rfl _ 1 rfl).mp hz | |
| apply exceptionalHom_not_boundary q hq (by simpa using h 1 (by decide)) | |
| refine ⟨f.hom, ?_⟩ | |
| have hh := congrArg ModuleCat.Hom.hom hf | |
| change f.hom.comp (dEll q (q ^ 0)) = exceptionalHom q at hh | |
| simpa only [pow_zero] using! hh | |
| end ArExplicit.Resolution | |
| end | |
| end OAI | |