| import Mathlib.Data.Finsupp.Weight |
| import Mathlib.LinearAlgebra.Dimension.Constructions |
| import Mathlib.LinearAlgebra.Dimension.DivisionRing |
| import Mathlib.LinearAlgebra.Dimension.RankNullity |
| import Mathlib.LinearAlgebra.Finsupp.Supported |
| import Mathlib.RingTheory.MvPolynomial.Homogeneous |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W64 |
|
|
| variable {k σ : Type*} [Field k] [Finite σ] |
|
|
| private theorem finite_exponents_of_degree (n : ℕ) : |
| {d : σ →₀ ℕ | d.degree = n}.Finite := |
| (Finsupp.finite_of_degree_le n).subset (fun _ h => le_of_eq h) |
|
|
| noncomputable instance homogeneousSection_finite (n : ℕ) : |
| Module.Finite k (MvPolynomial.homogeneousSubmodule σ k n) := by |
| rw [MvPolynomial.homogeneousSubmodule_eq_finsupp_supported] |
| let : Fintype {d : σ →₀ ℕ | d.degree = n} := |
| (finite_exponents_of_degree n).fintype |
| exact Module.Finite.equiv |
| (AddMonoidAlgebra.supportedEquivFinsupp (R := k) (S := k) |
| {d : σ →₀ ℕ | d.degree = n}).symm |
|
|
| theorem homogeneousSection_finrank (n : ℕ) : |
| Module.finrank k (MvPolynomial.homogeneousSubmodule σ k n) = |
| Nat.card {d : σ →₀ ℕ | d.degree = n} := by |
| rw [MvPolynomial.homogeneousSubmodule_eq_finsupp_supported] |
| let : Fintype {d : σ →₀ ℕ | d.degree = n} := |
| (finite_exponents_of_degree n).fintype |
| calc |
| _ = Module.finrank k ({d : σ →₀ ℕ | d.degree = n} →₀ k) := |
| (AddMonoidAlgebra.supportedEquivFinsupp (R := k) (S := k) |
| {d : σ →₀ ℕ | d.degree = n}).finrank_eq |
| _ = Nat.card {d : σ →₀ ℕ | d.degree = n} := by |
| simp only [Module.finrank_finsupp_self, Nat.card_eq_fintype_card] |
|
|
| noncomputable def homogeneousMultiply {d : ℕ} (f : MvPolynomial σ k) |
| (hf : f.IsHomogeneous d) (n : ℕ) : |
| MvPolynomial.homogeneousSubmodule σ k n →ₗ[k] |
| MvPolynomial.homogeneousSubmodule σ k (n + d) where |
| toFun x := ⟨x.1 * f, x.2.mul hf⟩ |
| map_add' x y := by apply Subtype.ext; exact add_mul _ _ _ |
| map_smul' c x := by apply Subtype.ext; exact smul_mul_assoc _ _ _ |
| |
| omit [Finite σ] in |
| theorem homogeneousMultiply_injective {d : ℕ} (f : MvPolynomial σ k) |
| (hf : f.IsHomogeneous d) (hf0 : f ≠ 0) (n : ℕ) : |
| Function.Injective (homogeneousMultiply f hf n) := by |
| intro x y h |
| apply Subtype.ext |
| exact mul_right_cancel₀ hf0 (congrArg Subtype.val h) |
|
|
| theorem homogeneousCokernel_finrank_add {d : ℕ} (f : MvPolynomial σ k) |
| (hf : f.IsHomogeneous d) (hf0 : f ≠ 0) (n : ℕ) : |
| Module.finrank k |
| (MvPolynomial.homogeneousSubmodule σ k (n + d) ⧸ |
| LinearMap.range (homogeneousMultiply f hf n)) + |
| Module.finrank k (MvPolynomial.homogeneousSubmodule σ k n) = |
| Module.finrank k (MvPolynomial.homogeneousSubmodule σ k (n + d)) := by |
| have hi := homogeneousMultiply_injective f hf hf0 n |
| simpa only [LinearMap.finrank_range_of_inj hi] using |
| (LinearMap.range (homogeneousMultiply f hf n)).finrank_quotient_add_finrank |
|
|
| end PiExponentJets.W64 |
|
|
| end OAI |
|
|
