regular local hilbert samuel multiplicity one

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

associated graded ring of a regular local ring: If $(R,\mathfrak{m},k)$ is regular local of dimension $d$, any cotangent basis induces a graded isomorphism $k[X_1,\dots ,X_d]\cong {\mathrm{gr}}_{\mathfrak{m}}R$. Conversely, if the associated graded ring of a nonzero Noetherian local ring is isomorphic as a graded $k$-algebra to $k[X_1,\dots ,X_d]$ with standard grading, then $R$ is regular of dimension $d$.

Hilbert-Samuel multiplicity as the factorial-scaled leading coefficient: Let $(R,\mathfrak{m})$ be a Noetherian local ring, let $M$ be a finite $R$-module, and let $I$ be an ideal of definition for $M$. If $M=0$, define $e_I(M):=0.$ If $M\mathrm{\ne }0$, let $P_{I,M}$ be the eventual Hilbert-Samuel polynomial from thm-existence-of-hilbert-samuel-polynomial, and let $d=\mathrm{deg}{P}_{I,M}$. Because $I\subseteq \mathfrak{m}$ and $M\mathrm{\ne }0$, Nakayama's lemma makes $M\mathrm{/}{I}^{n+1}M$ nonzero for every $n$, so $P_{I,M}$ is not the zero polynomial and $d$ is defined. The Hilbert-Samuel multiplicity of $M$ with respect to $I$ is $e_I(M):=d!\cdot (\text{leading coefficient of }{P}_{I,M})\mathrm{.}$ Equivalently, when $M\mathrm{\ne }0$ and $P_{I,M}(n)=\frac{e_I(M)}{d!}{n}^d+\text{lower-degree terms},$ then $e_I(M)$ is the integer scaling the top term.