File size: 2,970 Bytes
b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | 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
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