Derivative ideals under a multiple test blow-up

The centers are admissible for the derivative ideal. Let $0\le j\le \mu$ and let $({\mathcal{D}}^j(\mathcal{I},\mu ){)}_i$ denote the $i$-th controlled transform of the marked ideal ${\mathcal{D}}^j(\mathcal{I},\mu )$ along the same sequence. We prove by induction on $i$ that the sequence $(X_i)$ is a multiple test blow-up of ${\mathcal{D}}^j(\mathcal{I},\mu )$ and that $({\mathcal{D}}^j(\mathcal{I},\mu ){)}_i\subseteq {\mathcal{D}}^j({\mathcal{I}}_i,\mu )$. For $i=0$ this is equality. Assume it for $i$. By [F2] applied at stage $i$, $C_i\subseteq \mathrm{supp}({\mathcal{I}}_i,\mu )\subseteq \mathrm{supp}({\mathcal{D}}^j({\mathcal{I}}_i),\mu -j)$, and the latter is contained in $\mathrm{supp}(({\mathcal{D}}^j(\mathcal{I},\mu ){)}_i,\mu -j)$ by the induction inclusion (a smaller ideal has a larger order-superlevel support), so $C_i$ is an admissible center for the derivative marked ideal and step $i+1$ is defined for it.

The transform inclusion. With the notation of step 1.1, $({\mathcal{D}}^j(\mathcal{I},\mu ){)}_{i+1}={\sigma }_{i+1}^{\mathrm{c}}(({\mathcal{D}}^j(\mathcal{I},\mu ){)}_i)\subseteq {\sigma }_{i+1}^{\mathrm{c}}({\mathcal{D}}^j({\mathcal{I}}_i,\mu ))\subseteq {\mathcal{D}}^j({\sigma }_{i+1}^{\mathrm{c}}({\mathcal{I}}_i,\mu ))={\mathcal{D}}^j({\mathcal{I}}_{i+1},\mu )$, using the induction inclusion and the monotonicity of the controlled transform in step 1, and [F3] in step 2. This completes the induction and proves both assertions.