| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| namespace PiExponent |
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| open DirectSum |
| |
| variable {A : Type*} [CommRing A] |
| (𝒜 : ℕ → Submodule ℤ A) [GradedAlgebra 𝒜] |
| (S : Subring A) |
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| def subringGrade (n : ℕ) : Submodule ℤ S := |
| (𝒜 n).comap S.subtype.toAddMonoidHom.toIntLinearMap |
| |
| theorem subringGrade_independent : iSupIndep (subringGrade 𝒜 S) := by |
| have h := (DirectSum.Decomposition.isInternal 𝒜).submodule_iSupIndep |
| intro i |
| rw [disjoint_iff_inf_le] |
| intro x hx |
| apply Subtype.ext |
| apply (disjoint_iff_inf_le.mp (h i)) |
| refine ⟨hx.1, ?_⟩ |
| have hle : (⨆ j, ⨆ (_ : j ≠ i), subringGrade 𝒜 S j) ≤ |
| (⨆ j, ⨆ (_ : j ≠ i), 𝒜 j).comap S.subtype.toAddMonoidHom.toIntLinearMap := by |
| refine iSup_le fun j => iSup_le fun hj => ?_ |
| exact Submodule.comap_mono (le_iSup_of_le j (le_iSup_of_le hj le_rfl)) |
| exact hle hx.2 |
|
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| theorem subringGrade_spans |
| (hS : ∀ n x, x ∈ S → (decompose 𝒜 x n : A) ∈ S) : |
| (⨆ n, subringGrade 𝒜 S n) = ⊤ := by |
| classical |
| apply top_unique |
| intro x _ |
| let c : ℕ → S := fun n => ⟨decompose 𝒜 x.1 n, hS n x.1 x.2⟩ |
| have he : x = ∑ n ∈ (decompose 𝒜 x.1).support, c n := by |
| apply Subtype.ext |
| change x.1 = S.subtype (∑ n ∈ (decompose 𝒜 x.1).support, c n) |
| rw [map_sum] |
| exact (sum_support_decompose 𝒜 x.1).symm |
| rw [he] |
| apply Submodule.sum_mem |
| intro n hn |
| exact (le_iSup (subringGrade 𝒜 S) n) (show c n ∈ subringGrade 𝒜 S n from |
| (decompose 𝒜 x.1 n).2) |
|
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| @[instance_reducible] |
| def subringGradedAlgebra |
| (hS : ∀ n x, x ∈ S → (decompose 𝒜 x n : A) ∈ S) : |
| GradedAlgebra (subringGrade 𝒜 S) where |
| one_mem := show (1 : A) ∈ 𝒜 0 from SetLike.one_mem_graded 𝒜 |
| mul_mem := by |
| intro i j x y hx hy |
| change (x.1 * y.1) ∈ 𝒜 (i + j) |
| exact SetLike.mul_mem_graded hx hy |
| toDecomposition := |
| (DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top |
| (subringGrade_independent 𝒜 S) (subringGrade_spans 𝒜 S hS)).chooseDecomposition |
|
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| end PiExponent |
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| end |
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| end OAI |
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