Iterated derivative ideals preserve support in the safe characteristic range

Given: A marked ideal $(\mathcal{I},\mu )$ on a smooth $K$-scheme $X$ and an integer $0\le i\le \mu -1$.

Derivative ideals of an ideal sheaf and of a marked ideal: ${\mathcal{D}}^0(\mathcal{I})=\mathcal{I}$, ${\mathcal{D}}^i(\mathcal{I})$ is generated locally by generators $f$ of $\mathcal{I}$ and their coordinate partial derivatives of order at most $i$; the recursive identity ${\mathcal{D}}^i({\mathcal{D}}^j(\mathcal{I}))={\mathcal{D}}^{i+j}(\mathcal{I})$ holds in every characteristic.

Order of an ideal sheaf at a point: ${\mathrm{ord}}_x(\mathcal{I})=\max \{n:{\mathcal{I}}_x\subseteq {\mathfrak{m}}_x^n\}$, with order at least $\mu$ equivalent to vanishing in ${\mathcal{O}}_{X,x}\mathrm{/}{\mathfrak{m}}_x^{\mu }$.

Marked ideals and their support: $\mathrm{supp}(\mathcal{I},\mu )=\{x:{\mathrm{ord}}_x(\mathcal{I})\ge \mu \}$.

The regular-local associated-graded and completion theorems identify ${\mathrm{gr}}_{\mathfrak{m}}A$ with $\kappa (x)[U_1,\dots ,U_d]$ and its completion with $\kappa (x)\mathrm{⟦}{u}_1,\dots ,u_d\mathrm{⟧}$ when $K$ is perfect. A coefficient field containing $K$ is obtained by lifting a separating transcendence basis of $\kappa (x)\mathrm{/}K$, then its finite separable algebraic generators by the coefficient-field adjunction lemmas. The parameter map is surjective by the Cohen presentation and injective by the associated-graded isomorphism. These are associated graded ring of a regular local ring, Completion of a Noetherian local ring is local with the same residue field, completion preserves regular local rings, A complete equicharacteristic Noetherian local ring is a power-series quotient, Transcendental residue elements adjoin across a maximal subfield, and Separable residue elements adjoin across a maximal subfield. Formal parameter derivatives restrict to derivations $A\to \widehat{A}$; since ${\mathrm{\Omega }}_{A\mathrm{/}K}$ is finite free locally, universality identifies them with $\widehat{A}$-linear combinations of the algebraic derivations (Differentials of a smooth morphism, Derivations are maps out of Ω). Iterated Leibniz therefore puts their order-$r$ derivatives of $f\in I$ in ${\mathcal{D}}^r(I)\widehat{A}$.

On an étale chart to affine $N$-space, the infinitesimal Taylor map with coordinate increments $t_1,\dots ,t_N$ exists uniquely by Etale morphisms are the formally etale morphisms locally of finite presentation, modulo $(t{)}^{\mu }$. Its finitely many coefficients for a function $f$ are the Hasse derivatives of orders $<\mu$, regular functions on the chart. Over a perfect field, their residues all vanish at $x$ exactly when $f\in {\mathfrak{m}}_x^{\mu }$. Indeed the completed chart can be expressed using a separating residue-field coordinate system and the normal parameters in [F4]; Taylor substitution in the normal parameters detects every nonzero initial form of degree $<\mu$. An invertible change of smooth coordinates gives invertible changes of these truncated Taylor coefficients. This reasoning concerns Hasse derivatives, and uses no factorial division.