Order of an ideal sheaf at a point

Let $X$ be a locally Noetherian scheme, $\mathcal{I}\subseteq {\mathcal{O}}_X$ a coherent ideal sheaf (Coherent module sheaves, Ideal sheaves) and $x\in X$ with maximal ideal ${\mathfrak{m}}_x\subset {\mathcal{O}}_{X,x}$ (A local ring is a nonzero commutative ring with a unique maximal ideal). The order of $\mathcal{I}$ at $x$ is ${\mathrm{ord}}_x(\mathcal{I}):=\max \{n\ge 0:{\mathcal{I}}_x\subseteq {\mathfrak{m}}_x^n\},$ with ${\mathrm{ord}}_x(\mathcal{I}):=+\mathrm{\infty }$ when ${\mathcal{I}}_x=0$; the maximum is attained for ${\mathcal{I}}_x\mathrm{\ne }0$ because ${\bigcap }_n{\mathfrak{m}}_x^n=0$ in the Noetherian local ring ${\mathcal{O}}_{X,x}$ (The Krull intersection is the $(1-a)$-torsion submodule, and it vanishes in the Jacobson-radical case). For $f\in {\mathcal{O}}_X(U)$ put ${\mathrm{ord}}_x(f):={\mathrm{ord}}_x(f{\mathcal{O}}_U)$; a germ is of multiplicity one at $x$ if ${\mathrm{ord}}_x(f)=1$. If in addition ${\mathcal{O}}_{X,x}$ is regular — the only case used on this page — then $f$ is part of a regular system of parameters of ${\mathcal{O}}_{X,x}$, so the zero scheme $V(f)$ is regular at $x$ of dimension $\dim {\mathcal{O}}_{X,x}-1$ (regular local quotient by parameter is regular, under AC The Axiom of Choice). The parameter assertion follows by extending the nonzero class of $f$ in the finite-dimensional cotangent space to a basis and applying the finite-generator Nakayama argument. If $x\in V(\mathcal{I})$ and $s=\dim {\mathcal{O}}_{X,x}$, then ${\mathrm{ord}}_x(\mathcal{I})\ge 1$; ${\mathrm{ord}}_x$ is the multiplicity used throughout this page, and it differs from the Hilbert-Samuel multiplicity, which is not used here.