regular local domain induction

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 $d$.

The Krull intersection is the $(1-a)$-torsion submodule, and it vanishes in the Jacobson-radical case: The first clause below is choice-free; the second uses the published Jacobson-radical unit criterion and therefore inherits its Axiom-of-Choice boundary. Let $R$ be a Noetherian commutative ring, let $I\subset R$ be an ideal, and let $M$ be a finite $R$-module. Put $K:={\bigcap }_{n\ge 0}{I}^{n}M\mathrm{.}$ Then: 1. $K$ is exactly the set of elements $m\in M$ for which $(1-a)m=0$ for some $a\in I$; 2. if $I\subseteq J(R)$, then $K=0$.