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Derivative ideals of a maximal-order marked ideal have maximal order

Statement

Assume $μ \geq 1$ and either $char K = 0$ or $K$ perfect with $char K = p > μ$ (Field). If $(I, E, μ)$ is a marked ideal of maximal order, then for every $0 \leq i \leq μ$ , the derivative marked ideal $D^{i} (I, μ) = (D^{i} (I), μ - i)$ is of maximal order (Derivative ideals of an ideal sheaf and of a marked ideal).

Facts & Assumptions

Given: A field $K$ , a maximal-order marked ideal $(I, E, μ)$ with $μ \geq 1$ , and either $char K = 0$ or $K$ perfect with $char K = p > μ$ .

[F1]

Marked ideals of maximal order, tangent directions and transversality to the exceptional divisors: In characteristic zero or perfect characteristic $p > ν$ , for a marking $ν \geq 1$ , maximal order is equivalent to $D^{ν} (A) = O_{X}$ .

[F2]

Derivative ideals of an ideal sheaf and of a marked ideal: $D^{i} (I, μ) = (D^{i} (I), μ - i)$ and the recursive definition gives $D^{a} (D^{b} (I)) = D^{a + b} (I)$ in every characteristic.

Proof

1.1F1F2given

By maximality and the safe-characteristic hypothesis, $D^{μ} (I) = O_{X}$ by [F1]. For $0 \leq i \leq μ$ , the recursive identity in [F2] gives $D^{μ - i} (D^{i} (I)) = D^{μ} (I) = O_{X}$ .

2.1F1F2step 1.1∎

If $μ - i \geq 1$ , the field remains in characteristic zero or has $p > μ \geq μ - i$ , so [F1] implies that $D^{i} (I, μ)$ is of maximal order. If $i = μ$ , its residual marking is zero and its ideal is $O_{X}$ , which is maximal order directly. This proves the assertion for every $i$ .

Depends on

Used by

Canonical resolution of marked ideals Proposition

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jaroslaw Wlodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005) 779-822; author's arXiv version math/0401401 (28 pp., dated October 25, 2018) (standard reference, not scraped)