The maximal coefficient ideal of an ideal sheaf #
Kollár's maximal coefficient ideals [Kol07, Definition 98]: for a smooth variety X, an ideal sheaf
I ⊂ 𝒪_X and m = max-ord I, the maximal coefficient ideal of order s of I is
W_s(I) := ∑_{c : ∑ (m − j) c_j ≥ s} ∏_{j=0}^m (D^j I)^{c_j}, where D^j I is the j-th
derivative ideal sheaf. The same definition on a regular local ring with coordinates is
IsLocalRing.RegularCoords.W, with the weight IsLocalRing.wt m c = ∑ (m − j) c_j; Theorem
100 and Corollary 101 (Hironaka.Resolution.Algebraic.Tuning.Assemble,
Hironaka.Resolution.Algebraic.Tuning.Corollary101) speak of ideal sheaves, so the definition is
given here for ideal sheaves, with the parameter m explicit as in the maximal-contact ideal MC f I m (Kollár's m = max-ord I is a hypothesis where the theorems need it).
W f I m s := ⨆ (e : Fin (m + 1) → ℕ) (_ : s ≤ wt m e), ∏ j, (D^j I)^{e j}on Mathlib'sIdealSheafData(a complete lattice with products), with the characterisationsprod_le_W,W_le_iffand the unfoldingW_eq.- The elementary facts of [Kol07, Proposition 99] at the level of ideal sheaves, proved as on a
ring:
W_zero(W_0(I) = 𝒪_X),pow_le_W(I^c ⊆ W_s(I)whenmc ≥ s, the inclusionI^{s'} ⊆ W_{ms'}(I)through which the proof of Corollary 101 invokes Theorem 100),MC_pow_le_W(MC(I)^s ⊆ W_s(I)),W_one(W_1(I) = MC(I), part of [Kol07, Proposition 99 (4)]) and the antitonicityW_anti([Kol07, Proposition 99 (1)]). - The stalk bridge
stalkIdeal_W: for coordinatescon the stalk𝒪_{X,p}of a smooth variety that arek-linear and span thek-derivations,(W_s(I))_p = W_{c,s}(I_p), so that everything proved in coordinates aboutW(the parts of Proposition 99 inHironaka.Local) is available pointwise; the direction of Theorem 100 that needsW(Hironaka.Resolution.Algebraic.Tuning.Sequence) uses it.
Definition 98 for ideal sheaves #
The maximal coefficient ideal of order s of the ideal sheaf I [Kol07, Definition 98],
with the parameter m (Kollár's max-ord I) explicit:
W_s(I) = ∑_{wt_m(e) ≥ s} ∏_{j ≤ m} (D^j I)^{e_j}, the sum over all exponent vectors
e : {0, …, m} → ℕ of weight wt_m(e) = ∑ (m − j) e_j ≥ s, where D^j I is the j-th derivative
ideal sheaf derivativeIter f j I.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Definition 98 unfolded (definitional).
Every generating product of weight ≥ s lies in W_s(I) [Kol07, Definition 98].
W_s(I) ⊆ J iff every generating product of weight ≥ s lies in J [Kol07, Definition 98].
I^c ⊆ W_s(I) whenever mc ≥ s: the exponent vector e_0 = c has weight mc. This is the
inclusion I^{s'} ⊆ W_{ms'}(I) through which the proof of [Kol07, Corollary 101] invokes Theorem
100, and it is used again in the proof of [Kol07, Theorem 103].
MC(I)^s ⊆ W_s(I) for m ≥ 1 [Kol07, Proposition 99, proof]: the exponent vector e_{m−1} = s
has weight s.
W_1(I) = MC(I) for m ≥ 1
Kol07, Proposition 99 (4), proof: "W_1(I) ⊂ ∑_{j<m} D^j(I) = D^{m−1}(I)": ⊇ by MC_pow_le_W;
⊆ because a generating product of weight ≥ 1 has some e_j ≥ 1 with j < m, and so lies in
D^j I ⊆ D^{m−1} I.
W_t(I) ⊆ W_s(I) for s ≤ t [Kol07, Proposition 99 (1)].
The stalk bridge #
The stalk of the maximal coefficient ideal: for coordinates c on 𝒪_{X,p} that are k-linear
and span the k-derivations, (W_s(I))_p = W_{c,s}(I_p), the maximal coefficient ideal of the
stalk computed in coordinates (IsLocalRing.RegularCoords.W). The stalk of a sum of products of
powers is the sum of the products of the powers of the stalks, and (D^j I)_p = D_c^j(I_p)
(stalkIdeal_derivativeIter, derivativeIter_eq_Dpow).