Derivative ideals of an ideal sheaf and of a marked ideal

Assume AC (The Axiom of Choice) through the smooth-differential supplier. Let $X$ be a smooth $K$-scheme (Smooth morphism of schemes) with sheaf of relative differentials ${\mathrm{\Omega }}_{X\mathrm{/}K}$ (Sheaf of relative Kähler differentials, Existence and generators of Kähler differentials) and derivation sheaf ${\mathrm{Der}}_K({\mathcal{O}}_X)$ (Derivation of an algebra, Derivations are maps out of Ω). For a coherent ideal sheaf $\mathcal{I}\subseteq {\mathcal{O}}_X$ (Coherent module sheaves) define the first derivative (extension) $\mathcal{D}(\mathcal{I})$ intrinsically as the ideal generated by $\mathcal{I}$ and all $D(f)$, for local sections $f$ of $\mathcal{I}$ and $K$-derivations $D$. The smooth-differential theorem Differentials of a smooth morphism makes ${\mathrm{\Omega }}_{X\mathrm{/}K}$ finite locally free. Choose functions $u_1,\dots ,u_n$ whose differentials form a basis on a neighbourhood and let $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{u}_i$ be the dual derivations. Here $n$ is the relative dimension over $K$, which need not equal the local-ring dimension at a nonclosed point. For generators $f_1,\dots ,f_s$ of $\mathcal{I}$, the same ideal is generated by the $f_j$ and $\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{u}_i$: Leibniz reduces derivatives of arbitrary $\sum a_j{f}_j$ to these generators, and every derivation is a linear combination of the dual basis. This finite generator list proves coherence and independence of generators and differential coordinates. Set ${\mathcal{D}}^0(\mathcal{I}):=\mathcal{I}$ and ${\mathcal{D}}^i(\mathcal{I}):=\mathcal{D}({\mathcal{D}}^{i-1}(\mathcal{I}))$. The ideal is generated by $f_j$ and their derivatives of order at most $i$ and is independent of the chosen generators and coordinates; for a marked ideal put ${\mathcal{D}}^i(\mathcal{I},\mu ):=({\mathcal{D}}^i(\mathcal{I}),\mu -i)$ for $0\le i\le \mu$. In characteristic $p>0$, ordinary derivations need not lower order: on ${\mathbb{A}}_K^1$ for a field $K$ of characteristic $p$, the ideal $\mathcal{I}=(x^p)$ satisfies $\mathcal{D}(\mathcal{I})=\mathcal{I}$.