Controlled derivative transforms are contained in derivatives of the controlled transform

Given: A marked ideal $(\mathcal{I},\mu )$ on a smooth $K$-scheme $X$, a regular center $C\subseteq \mathrm{supp}(\mathcal{I},\mu )$ with SNC with $E$, the blowup $\sigma \text{\hspace{0.17em}⁣}:X^{\mathrm{\prime }}\to X$ with exceptional divisor $D$ and local equation $y$, and $0\le r\le \mu$.

Derivative ideals of an ideal sheaf and of a marked ideal: $\mathcal{D}(\mathcal{A})$ is generated by the local sections of $\mathcal{A}$ and their first derivatives with respect to local coordinates; equivalently by the sections $f\in \mathcal{A}$ together with $D(f)$ for $K$-derivations $D$ of ${\mathcal{O}}_X$; ${\mathcal{D}}^r$ is the iterate and ${\mathcal{D}}^r(\mathcal{I},\mu )=({\mathcal{D}}^r(\mathcal{I}),\mu -r)$.

Multiple test blow-ups, controlled transforms and resolutions of marked ideals, Controlled transforms are well defined: the controlled transform is ${\sigma }^{\mathrm{c}}(\mathcal{A},\nu )=(\mathcal{I}(D{)}^{-\nu }{\sigma }^{\ast }\mathcal{A},\nu )$, computed by $f\mapsto y^{-\nu }{\sigma }^{\ast }(f)$ on sections; it is an ideal sheaf, well defined up to units.

embedding dimension and regular local ring, Blowup of a scheme along an ideal sheaf: at a point of $C$ choose local parameters $u_1,\dots ,u_n$ with $C$ described by $u_1=\cdots =u_m=0$ and $y=u_m$; the corresponding chart of the blowup has coordinates $u_i^{\mathrm{\prime }}=u_i\mathrm{/}y$ $(i<m)$, $u_i^{\mathrm{\prime }}=u_i$ $(i>m)$, $u_m^{\mathrm{\prime }}=u_m=y$.

Derivation of an algebra, Derivations are maps out of Ω: the $K$-derivations of the structure sheaf form a locally free sheaf; a derivation is determined by its values on local coordinates, and $y{\sigma }^{\ast }(D)$ for a derivation $D$ on $X$ acts on ${\sigma }^{\ast }(f)$ by $y{\sigma }^{\ast }(Df)$.

Iterated derivative ideals preserve support in the safe characteristic range: for $0\le j\le \mu -1$, $\mathrm{supp}(\mathcal{I},\mu )\subseteq \mathrm{supp}({\mathcal{D}}^j(\mathcal{I}),\mu -j)$ in every characteristic, so the center hypothesis transfers to the lower derivative ideal.