associated graded ring of a regular local ring

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 polynomial surjection: Let $(R,\mathfrak{m},k)$ be nonzero Noetherian local and let $x_1,\dots ,x_e$ lift a basis of $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$. There is a surjective graded $k$-algebra map $\varphi :k[X_1,\dots ,X_e]\to {\mathrm{gr}}_{\mathfrak{m}}R$, determined by $X_i\mapsto x_i+{\mathfrak{m}}^2$, with every variable of degree one.

regular local graded surjection has zero kernel: Assume the Axiom of Choice. For the graded map $\varphi :k[X_1,\dots ,X_e]\twoheadrightarrow {\mathrm{gr}}_{\mathfrak{m}}R$ defined by a cotangent basis in a nonzero Noetherian local ring, if $e=\dim R$, then $\ker \varphi =0$.

The degree of the Hilbert-Samuel polynomial equals the dimension of the support: Assume the Axiom of Choice. Let $(R,\mathfrak{m})$ be a Noetherian local ring, let $M\mathrm{\ne }0$ be a finite $R$-module, and let $I$ be an ideal of definition for $M$. Then the Hilbert-Samuel polynomial $P_{I,M}$ has degree $\mathrm{deg}{P}_{I,M}=\dim \mathrm{Supp}(M)\mathrm{.}$