| import OAI.Algebra.AuslanderReiten.Evaluation.TensorBeta |
| import OAI.Algebra.AuslanderReiten.Stable.StableHomEquiv |
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| namespace OAI |
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| noncomputable section |
| namespace ArExplicit.A |
| universe u |
| variable {k : Type u} [Field k] (q : k) (hq1 : 1 + q ≠ 0) |
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| include hq1 in |
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| theorem simple_not_projective : ¬ Module.Projective (A k q) (Simple q) := by |
| intro hp |
| let := hp |
| let : Module.Finite (A k q) (Simple q) := Module.Finite.of_restrictScalars_finite k (A k q) _ |
| exact beta_stable_ne_zero q hq1 (StableHom.eq_zero_of_projective_source _) |
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|
| include hq1 in |
| theorem simple_scalar_generator (x : Simple q) : |
| x = simpleCoordinate q x • simpleGenerator q := by |
| apply simpleCoordinate_injective q hq1 |
| erw [(simpleCoordinate q).map_smul, simpleGenerator_coordinate] |
| exact (mul_one _).symm |
|
|
| def endScalar : (Simple q →ₗ[A k q] Simple q) →ₗ[k] k where |
| toFun f := simpleCoordinate q (f (simpleGenerator q)) |
| map_add' left right := |
| (simpleCoordinate q).map_add (left (simpleGenerator q)) (right (simpleGenerator q)) |
| map_smul' scalar endomorphism := |
| (simpleCoordinate q).map_smul scalar (endomorphism (simpleGenerator q)) |
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|
| include hq1 in |
| theorem end_eq_scalar_id (f : Simple q →ₗ[A k q] Simple q) : |
| f = endScalar q f • LinearMap.id := by |
| apply LinearMap.ext |
| intro x |
| change f x = endScalar q f • x |
| calc |
| f x = f (simpleCoordinate q x • simpleGenerator q) := congrArg f (simple_scalar_generator q hq1 x) |
| _ = simpleCoordinate q x • f (simpleGenerator q) := (f.restrictScalars k).map_smul _ _ |
| _ = simpleCoordinate q x • (endScalar q f • simpleGenerator q) := by |
| exact congrArg (fun z => simpleCoordinate q x • z) |
| (simple_scalar_generator q hq1 (f (simpleGenerator q))) |
| _ = endScalar q f • (simpleCoordinate q x • simpleGenerator q) := smul_comm _ _ _ |
| _ = endScalar q f • x := congrArg (fun z => endScalar q f • z) (simple_scalar_generator q hq1 x).symm |
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| def simpleEndEquiv : (Simple q →ₗ[A k q] Simple q) ≃ₗ[k] k where |
| __ := endScalar q |
| invFun c := c • LinearMap.id |
| left_inv f := (end_eq_scalar_id q hq1 f).symm |
| right_inv c := by |
| change simpleCoordinate q (c • simpleGenerator q) = c |
| erw [(simpleCoordinate q).map_smul, simpleGenerator_coordinate] |
| exact mul_one _ |
|
|
| include hq1 in |
| theorem stable_identity_ne_zero : |
| StableHom.classOf (A k q) (Simple q) (Simple q) LinearMap.id ≠ 0 := by |
| let : Module.Finite (A k q) (Simple q) := Module.Finite.of_restrictScalars_finite k (A k q) _ |
| intro h |
| exact simple_not_projective q hq1 (StableHom.identity_eq_zero_iff.mp h) |
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| include hq1 in |
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| theorem simpleEnd_classOf_injective : Function.Injective |
| (StableHom.classOfLinear k : (Simple q →ₗ[A k q] Simple q) →ₗ[k] |
| StableHom (A k q) (Simple q) (Simple q)) := by |
| rw [← LinearMap.ker_eq_bot] |
| apply le_antisymm _ bot_le |
| intro f hf |
| have hz : StableHom.classOf (A k q) (Simple q) (Simple q) f = 0 := hf |
| rw [end_eq_scalar_id q hq1 f, StableHom.classOf_smul] at hz |
| have hc : endScalar q f = 0 := (smul_eq_zero.mp hz).resolve_right (stable_identity_ne_zero q hq1) |
| rw [end_eq_scalar_id q hq1 f, hc, zero_smul] |
| exact Submodule.zero_mem _ |
|
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| def stableEndEquiv : StableHom (A k q) (Simple q) (Simple q) ≃ₗ[k] k := |
| (LinearEquiv.ofBijective (StableHom.classOfLinear k) |
| ⟨simpleEnd_classOf_injective q hq1, StableHom.classOf_surjective⟩).symm.trans |
| (simpleEndEquiv q hq1) |
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| @[simp] |
| theorem stableEndEquiv_classOf (f : Simple q →ₗ[A k q] Simple q) : |
| stableEndEquiv q hq1 (StableHom.classOf (A k q) _ _ f) = endScalar q f := by |
| change simpleEndEquiv q hq1 ((LinearEquiv.ofBijective (StableHom.classOfLinear k) _).symm |
| ((StableHom.classOfLinear k) f)) = _ |
| erw [LinearEquiv.symm_apply_apply] |
| rfl |
|
|
| @[simp] |
| theorem stableEndEquiv_identity : stableEndEquiv q hq1 StableHom.identity = 1 := by |
| change stableEndEquiv q hq1 (StableHom.classOf _ _ _ LinearMap.id) = 1 |
| rw [stableEndEquiv_classOf] |
| exact simpleGenerator_coordinate q |
|
|
| end ArExplicit.A |
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| end |
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| end OAI |
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