= Associated graded ring of a regular local ring
{title2=$\operatorname{gr}_{\mathfrak m}R\cong\kappa[T_1,\ldots,T_e]$}
For a <regular local ring> of <embedding dimension> $e$, a minimal maximal-<ideal> generating set induces the displayed polynomial-algebra isomorphism. It is surjective in every degree. A nonzero homogeneous kernel relation would bound the cumulative Hilbert function by $O(t^{e-1})$, contradicting the <prime-chain lower bound for local length> at $\dim R=e$. The <graded ring> is a domain; the <Krull intersection theorem> then shows the original ring is a domain by multiplying nonzero initial forms.
Back to article page
