| import OAI.Algebra.AuslanderReiten.Ext.BaseExt |
| import Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open CategoryTheory CategoryTheory.Abelian |
| open scoped ModuleCat.Algebra |
|
|
| namespace ArExplicit.Resolution |
|
|
| open ArExplicit.Letter ArExplicit.BaseAlgebra |
|
|
| variable {k : Type*} [Field k] [CharP k 2] (q : k) |
|
|
| local notation "C" => BaseAlgebra k q |
|
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| theorem dualHom_exceptional_decomposition (hq : q ≠ 0) (hq1 : 1 + q ≠ 0) |
| (g : E q →ₗ[C] C) (hg : g.comp (dEll q q) = 0) : |
| ∃ f : E q →ₗ[C] C, f.comp (dEll q 1) = |
| g - (g (vertexE q)).coeff v • exceptionalHom q := by |
| let r := (g (vertexE q)).coeff v |
| let g' := g - r • exceptionalHom q |
| have hg' : g'.comp (dEll q q) = 0 := by |
| simp [g', LinearMap.sub_comp, LinearMap.smul_comp, hg, exceptionalHom_cocycle] |
| have hc : leftEll q q (evaluateE q g') = 0 := by |
| rw [← evaluateE_precompose, hg'] |
| rfl |
| have hv : ((evaluateE q g' : rightE q) : C).coeff v = 0 := by |
| simp [evaluateE, g', r, exceptionalHom, fromRightE, vertexE, exceptionalCycle, |
| sub_eq_add_neg, basis_mul, multiplicationTable] |
| obtain ⟨a, ha⟩ := (exceptional_boundary_iff q hq hq1 _ hc).mpr hv |
| refine ⟨fromRightE q a, ?_⟩ |
| apply (homEEquiv q).injective |
| change evaluateE q ((fromRightE q a).comp (dEll q 1)) = evaluateE q g' |
| rw [evaluateE_precompose, evaluateE_fromRightE, ha] |
|
|
| omit [CharP k 2] in |
|
|
| theorem extMk_smul_regular {M : ModuleCat C} (P : ProjectiveResolution M) {n : ℕ} |
| (f : P.complex.X n ⟶ ModuleCat.of C C) (m : ℕ) (hm : n + 1 = m) |
| (hf : P.complex.d m n ≫ f = 0) (r : k) |
| (hrf : P.complex.d m n ≫ (r • f) = 0) : |
| P.extMk (r • f) m hm hrf = r • P.extMk f m hm hf := by |
| rw [Ext.smul_eq_comp_mk₀, P.extMk_comp_mk₀] |
| congr 1 |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro a |
| change r • f.hom a = algebraMap k C r * f.hom a |
| exact Algebra.smul_def r _ |
|
|
| def exceptionalExtLinear (h : NoPositivePowerOne q) : |
| k →ₗ[k] Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) 2 := |
| LinearMap.toSpanSingleton k _ (regular_ext_two_class q h) |
|
|
| theorem exceptionalExtLinear_injective (h : NoPositivePowerOne q) (hq : q ≠ 0) : |
| Function.Injective (exceptionalExtLinear q h) := by |
| intro a b hab |
| have heq : (a - b) • regular_ext_two_class q h = 0 := by |
| change a • regular_ext_two_class q h = b • regular_ext_two_class q h at hab |
| rw [sub_smul, hab, sub_self] |
| exact sub_eq_zero.mp ((smul_eq_zero.mp heq).resolve_right (regular_ext_two_class_ne_zero q h hq)) |
|
|
| theorem regular_ext_two_representative (h : NoPositivePowerOne q) (hq : q ≠ 0) |
| (g : (projectiveResolution q h).complex.X 2 ⟶ ModuleCat.of C C) |
| (hg : (projectiveResolution q h).complex.d 3 2 ≫ g = 0) : |
| (projectiveResolution q h).extMk g 3 rfl hg = |
| (g.hom (vertexE q)).coeff v • regular_ext_two_class q h := by |
| let P := projectiveResolution q h |
| have hg' : g.hom.comp (dEll q q) = 0 := by |
| have hh := congrArg ModuleCat.Hom.hom hg |
| change g.hom.comp (dEll q (q ^ 1)) = 0 at hh |
| simpa only [pow_one] using! hh |
| let r := (g.hom (vertexE q)).coeff v |
| obtain ⟨f, hf⟩ := dualHom_exceptional_decomposition q hq (by simpa using h 1 (by decide)) g.hom hg' |
| let vHom : P.complex.X 2 ⟶ ModuleCat.of C C := ModuleCat.ofHom (exceptionalHom q) |
| have hv : P.complex.d 3 2 ≫ vHom = 0 := by |
| apply ModuleCat.hom_ext |
| change (exceptionalHom q).comp (dEll q (q ^ 1)) = 0 |
| simpa only [pow_one] using exceptionalHom_cocycle q |
| have hrv : P.complex.d 3 2 ≫ (r • vHom) = 0 := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro b |
| have hb := congrArg (fun z => z.hom b) hv |
| change r • vHom.hom ((P.complex.d 3 2).hom b) = 0 |
| change vHom.hom ((P.complex.d 3 2).hom b) = 0 at hb |
| rw [hb, smul_zero] |
| have hd : P.complex.d 3 2 ≫ (g - r • vHom) = 0 := by |
| rw [Preadditive.comp_sub, hg, hrv, sub_zero] |
| have hbound : P.extMk (g - r • vHom) 3 rfl hd = 0 := by |
| apply (P.extMk_eq_zero_iff _ 3 rfl hd 1 rfl).mpr |
| refine ⟨ModuleCat.ofHom f, ?_⟩ |
| apply ModuleCat.hom_ext |
| change f.comp (dEll q (q ^ 0)) = (g - r • vHom).hom |
| apply LinearMap.ext |
| intro b |
| have hb := LinearMap.congr_fun hf b |
| simpa [r, vHom, pow_zero, Algebra.smul_def] using! hb |
| have hrep : P.extMk vHom 3 rfl hv = regular_ext_two_class q h := rfl |
| rw [← P.sub_extMk g (r • vHom) 3 rfl hg hrv, extMk_smul_regular q P vHom 3 rfl hv r hrv, hrep] at hbound |
| exact sub_eq_zero.mp hbound |
|
|
| theorem exceptionalExtLinear_surjective (h : NoPositivePowerOne q) (hq : q ≠ 0) : |
| Function.Surjective (exceptionalExtLinear q h) := by |
| intro a |
| obtain ⟨g, hg, rfl⟩ := (projectiveResolution q h).extMk_surjective a 3 rfl |
| exact ⟨(g.hom (vertexE q)).coeff v, (regular_ext_two_representative q h hq g hg).symm⟩ |
|
|
| def exceptionalExtEquiv (h : NoPositivePowerOne q) (hq : q ≠ 0) : |
| k ≃ₗ[k] Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) 2 := |
| LinearEquiv.ofBijective (exceptionalExtLinear q h) |
| ⟨exceptionalExtLinear_injective q h hq, exceptionalExtLinear_surjective q h hq⟩ |
|
|
| theorem regular_ext_two_finrank (h : NoPositivePowerOne q) (hq : q ≠ 0) : |
| Module.finrank k (Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) 2) = 1 := by |
| rw [← (exceptionalExtEquiv q h hq).finrank_eq] |
| exact Module.finrank_self k |
|
|
| theorem regular_ext_two_right_action (h : NoPositivePowerOne q) (hq : q ≠ 0) |
| (a : Ext (ModuleCat.of C (Simple q)) (ModuleCat.of C C) 2) (b : C) : |
| a.comp (Ext.mk₀ (ModuleCat.ofHom (rightMultiply q b))) (add_zero 2) = b.coeff f • a := by |
| let P := projectiveResolution q h |
| obtain ⟨g, hg, rfl⟩ := P.extMk_surjective a 3 rfl |
| have hgc : g.hom.comp (dEll q q) = 0 := by |
| have hh := congrArg ModuleCat.Hom.hom hg |
| change g.hom.comp (dEll q (q ^ 1)) = 0 at hh |
| simpa only [pow_one] using! hh |
| let gl : E q →ₗ[C] C := g.hom |
| have hgl : gl.comp (dEll q q) = 0 := hgc |
| have hc : leftEll q q (evaluateE q gl) = 0 := by |
| rw [← evaluateE_precompose, hgl] |
| rfl |
| rw [P.extMk_comp_mk₀] |
| rw [regular_ext_two_representative q h hq, regular_ext_two_representative q h hq] |
| change (g.hom (vertexE q) * b).coeff v • regular_ext_two_class q h = _ |
| have hcoeff := exceptional_right_action q hq (by simpa using h 1 (by decide)) |
| (evaluateE q gl) hc b |
| change (g.hom (vertexE q) * b).coeff v = (g.hom (vertexE q)).coeff v * b.coeff f at hcoeff |
| rw [hcoeff, mul_comm, mul_smul] |
|
|
| theorem exceptionalExtEquiv_right_action (h : NoPositivePowerOne q) (hq : q ≠ 0) |
| (r : k) (b : C) : |
| (exceptionalExtEquiv q h hq r).comp (Ext.mk₀ (ModuleCat.ofHom (rightMultiply q b))) |
| (add_zero 2) = exceptionalExtEquiv q h hq (r * b.coeff f) := by |
| rw [regular_ext_two_right_action q h hq] |
| change b.coeff f • (r • regular_ext_two_class q h) = (r * b.coeff f) • regular_ext_two_class q h |
| rw [← mul_smul, mul_comm] |
|
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| end ArExplicit.Resolution |
|
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| end |
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| end OAI |
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