@[reducible, inline]

abbrev AssocGradedPiece {A : Type u_1} [CommRing A] (m : Ideal A) (n : ℕ) : Type u_1

The n-th piece m^n / m^{n+1} of the associated graded ring of A with respect to an ideal m.

def AssocGraded {A : Type u_1} [CommRing A] (m : Ideal A) : Type u_1

The associated graded ring gr_m(A) = ⨁_n m^n/m^{n+1} of A with respect to m.

def imageInGradedPiece {A : Type u_1} [CommRing A] (m : Ideal A) (a : A) (p : ℕ) (ha : a ∈ m ^ p) : AssocGradedPiece m p

The image (initial form in degree p) of an element a ∈ m^p inside the graded piece m^p/m^{p+1}.

def gradedPieceToAssocGraded {A : Type u_1} [CommRing A] (m : Ideal A) (p : ℕ) (x : AssocGradedPiece m p) : AssocGraded m

Include a single graded piece m^p/m^{p+1} into the associated graded ring.

def imageInAssocGraded {A : Type u_1} [CommRing A] (m : Ideal A) (a : A) (p : ℕ) (ha : a ∈ m ^ p) : AssocGraded m

The image in the associated graded ring of an element a ∈ m^p, placed in degree p.

def quotientIdeal {A : Type u_1} [CommRing A] (m : Ideal A) (a : A) : Ideal (A ⧸ Ideal.span {a})

The image of an ideal m in the quotient ring A / (a).

def InitialFormNonZeroDivisor {A : Type u_1} [CommRing A] (m : Ideal A) (a : A) (p : ℕ) : Prop

The hypothesis used in Lemma 31/32 expressing that the initial form of a (an element of m^p) is a non-zero-divisor in the associated graded ring: if ax ∈ m^{k+p+1} and x ∈ m^k, then already x ∈ m^{k+1}.

theorem initialForm_nonZeroDivisor_intersection {A : Type u_1} [CommRing A] (m : Ideal A) (a : A) (p : ℕ) (hnd : InitialFormNonZeroDivisor m a p) (n : ℕ) (hn : p ≤ n) (c : A) (hac : a * c ∈ m ^ n) : c ∈ m ^ (n - p)

Iterated form of the non-zero-divisor hypothesis: if the initial form of a is regular in degree p, then a · c ∈ m^n (with n ≥ p) forces c ∈ m^{n-p}.

theorem lemma32_graded_quotient {A : Type u_1} [CommRing A] (m : Ideal A) (a : A) (p : ℕ) (ha : a ∈ m ^ p) (hnd : InitialFormNonZeroDivisor m a p) (n : ℕ) (hn : p ≤ n) : Ideal.span {a} ⊓ m ^ n = Ideal.span {a} * m ^ (n - p)

Lemma 32 (Lecture 19). If the initial form of a ∈ m^p is a non-zero-divisor in the associated graded ring, then for n ≥ p one has (a) ∩ m^n = (a) · m^{n-p}.