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: Assume the Axiom of Choice. 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 finite, and let $I\subseteq \mathfrak{m}$ be an ideal of definition for $M$. Define $e_I(0)=0$. For $M\mathrm{\ne }0$, when the Hilbert-Samuel function agrees eventually with a polynomial $P_{I,M}$, this polynomial is unique and nonzero; with $d=\mathrm{deg}{P}_{I,M}$, multiplicity is $e_I(M)=d!\cdot (\text{leading coefficient of }{P}_{I,M})$.