| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomLinear |
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| namespace OAI |
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| open CategoryTheory |
| open CochainComplex CochainComplex.HomComplex |
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| namespace ArExplicit.CompleteComparison |
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| universe u |
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| variable (k : Type u) [CommRing k] |
| variable {C : Type*} [Category* C] [Preadditive C] [Linear k C] |
| variable (K L : CochainComplex C ℤ) (n a n' : ℤ) |
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| noncomputable def rightShiftCocycleLinearEquiv (hn : n' + a = n) : |
| Cocycle K L n ≃ₗ[k] Cocycle K (L⟦a⟧) n' := |
| { Cocycle.rightShiftAddEquiv n a n' hn with |
| map_smul' := by |
| intro c x |
| apply Subtype.ext |
| exact Cochain.rightShift_smul x.val a n' hn c } |
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| variable {k K L n a n'} |
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| theorem rightShift_boundary (hn : n' + a = n) (x : Cocycle K L n) |
| (hx : x ∈ coboundaries K L n) : x.rightShift a n' hn ∈ coboundaries K (L⟦a⟧) n' := by |
| obtain ⟨m, hm, y, hy⟩ := hx |
| have hm' : n' - 1 + a = m := by omega |
| refine ⟨n' - 1, by omega, a.negOnePow • y.rightShift a (n' - 1) hm', ?_⟩ |
| rw [δ_units_smul, Cochain.δ_rightShift y a (n' - 1) n' hm' n hn, hy] |
| simp only [smul_smul, Int.units_mul_self, one_smul] |
| rfl |
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|
| theorem rightUnshift_boundary (hn : n' + a = n) (x : Cocycle K (L⟦a⟧) n') |
| (hx : x ∈ coboundaries K (L⟦a⟧) n') : x.rightUnshift n hn ∈ coboundaries K L n := by |
| obtain ⟨m', hm', y, hy⟩ := hx |
| have hm : m' + a = n - 1 := by omega |
| refine ⟨n - 1, by omega, a.negOnePow • y.rightUnshift (n - 1) hm, ?_⟩ |
| rw [δ_units_smul, Cochain.δ_rightUnshift y (n - 1) hm n n' hn, hy] |
| simp only [smul_smul, Int.units_mul_self, one_smul] |
| rfl |
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| variable (k K L n a n') |
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| noncomputable def rightShiftCohomologyEquiv (hn : n' + a = n) : |
| CohomologyClass K L n ≃ₗ[k] CohomologyClass K (L⟦a⟧) n' := |
| Submodule.Quotient.equiv (fullHomCoboundaries k K L n) |
| (fullHomCoboundaries k K (L⟦a⟧) n') (rightShiftCocycleLinearEquiv k K L n a n' hn) (by |
| ext y |
| constructor |
| · rintro ⟨x, hx, rfl⟩ |
| exact rightShift_boundary hn x hx |
| · intro hy |
| let E := rightShiftCocycleLinearEquiv k K L n a n' hn |
| refine ⟨E.symm y, rightUnshift_boundary hn y hy, E.apply_symm_apply y⟩) |
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| noncomputable def leftShiftCocycleLinearEquiv (hn : n + a = n') : |
| Cocycle K L n ≃ₗ[k] Cocycle (K⟦a⟧) L n' := |
| { Cocycle.leftShiftAddEquiv n a n' hn with |
| map_smul' := by |
| intro c x |
| apply Subtype.ext |
| exact Cochain.leftShift_smul x.val a n' hn c } |
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| variable {k K L n a n'} |
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| theorem leftShift_boundary (hn : n + a = n') (x : Cocycle K L n) |
| (hx : x ∈ coboundaries K L n) : x.leftShift a n' hn ∈ coboundaries (K⟦a⟧) L n' := by |
| obtain ⟨m, hm, y, hy⟩ := hx |
| have hm' : m + a = n' - 1 := by omega |
| refine ⟨n' - 1, by omega, a.negOnePow • y.leftShift a (n' - 1) hm', ?_⟩ |
| rw [δ_units_smul, Cochain.δ_leftShift y a (n' - 1) n' hm' n hn, hy] |
| simp only [smul_smul, Int.units_mul_self, one_smul] |
| rfl |
|
|
| theorem leftUnshift_boundary (hn : n + a = n') (x : Cocycle (K⟦a⟧) L n') |
| (hx : x ∈ coboundaries (K⟦a⟧) L n') : x.leftUnshift n hn ∈ coboundaries K L n := by |
| obtain ⟨m', hm', y, hy⟩ := hx |
| have hm : n - 1 + a = m' := by omega |
| refine ⟨n - 1, by omega, a.negOnePow • y.leftUnshift (n - 1) hm, ?_⟩ |
| rw [δ_units_smul, Cochain.δ_leftUnshift y (n - 1) hm n n' hn, hy] |
| simp only [smul_smul, Int.units_mul_self, one_smul] |
| rfl |
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| variable (k K L n a n') |
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| noncomputable def leftShiftCohomologyEquiv (hn : n + a = n') : |
| CohomologyClass K L n ≃ₗ[k] CohomologyClass (K⟦a⟧) L n' := |
| Submodule.Quotient.equiv (fullHomCoboundaries k K L n) |
| (fullHomCoboundaries k (K⟦a⟧) L n') (leftShiftCocycleLinearEquiv k K L n a n' hn) (by |
| ext y |
| constructor |
| · rintro ⟨x, hx, rfl⟩ |
| exact leftShift_boundary hn x hx |
| · intro hy |
| let E := leftShiftCocycleLinearEquiv k K L n a n' hn |
| refine ⟨E.symm y, leftUnshift_boundary hn y hy, E.apply_symm_apply y⟩) |
|
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| end ArExplicit.CompleteComparison |
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| end OAI |
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