Controlled transforms are well defined

Given: A marked ideal $(\mathcal{I},E,\mu )$ on a smooth $K$-scheme $X$, a regular closed subscheme $C\subseteq \mathrm{supp}(\mathcal{I},E,\mu )$ with SNC with $E$, the blowup $\sigma \text{\hspace{0.17em}⁣}:X^{\mathrm{\prime }}\to X$ of $C$ with exceptional divisor $D$, the ideal ${\mathcal{I}}_C$ of $C$ and the invertible ideal $\mathcal{I}(D)$ of $D$.

Marked ideals and their support: the support is $\mathrm{supp}(\mathcal{I},E,\mu )=\{x:{\mathrm{ord}}_x(\mathcal{I})\ge \mu \}$, and $C\subseteq \mathrm{supp}(\mathcal{I},E,\mu )$ means ${\mathrm{ord}}_x(\mathcal{I})\ge \mu$ for every $x\in C$.

Order of an ideal sheaf at a point: ${\mathrm{ord}}_x(\mathcal{I})=\max \{n:{\mathcal{I}}_x\subseteq {\mathfrak{m}}_x^n\}$; equivalently, ${\mathrm{ord}}_x(\mathcal{I})$ is the minimum of ${\mathrm{ord}}_x(f)$ over local sections $f$ of $\mathcal{I}$ at $x$.

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier, Exceptional subscheme of a blowup, Invertible sheaf of cartier divisor: ${\sigma }^{\ast }{\mathcal{I}}_C=\mathcal{I}(D)$ is the invertible ideal of the exceptional divisor, generated locally by the equation $y$ of $D$.

Multiple test blow-ups, controlled transforms and resolutions of marked ideals: the controlled transform of $(\mathcal{I},\mu )$ along $\sigma$ is ${\sigma }^{\mathrm{c}}(\mathcal{I},\mu )=(\mathcal{I}(D{)}^{-\mu }{\sigma }^{\ast }\mathcal{I},\mu )$; for a local section $f$ the section $y^{-\mu }{\sigma }^{\ast }(f)$ is called a controlled transform of $f$.

regular local regular quotient ideal is parameter generated, regular local rings are domains and cohen macaulay: at a point $x\in C$ the ideal ${\mathcal{I}}_C$ is generated by parameters $u_1,\dots ,u_k$ that extend to a regular system of parameters $u_1,\dots ,u_n$ of ${\mathcal{O}}_{X,x}$; in particular $C$ is reduced at $x$, and a section not in ${\mathcal{I}}_C$ has a nonvanishing value at some point of $C$.

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier, Effective cartier divisor: the exceptional divisor $D$ is an effective Cartier divisor with invertible ideal $\mathcal{I}(D)$ and local equation $y$ a nonzerodivisor.

Associated graded algebra of an ideal generated by a regular sequence (Stacks, Lemma 10.69.2): if $P=(u_1,\dots ,u_c)$ is generated by a regular sequence in $R$, then ${\mathrm{gr}}_{P}R=(R\mathrm{/}P)[U_1,\dots ,U_c]$. In particular each $P^j\mathrm{/}{P}^{j+1}$ is a finite free $R\mathrm{/}P$-module. The proof eliminates a homogeneous relation by induction on the sequence length and its degree, using nonzerodivisibility of the last generator modulo the preceding ones.