Regular local ring

A Noetherian local ring $(R,\mathfrak{m})$ is regular when the minimal number of generators of its maximal ideal equals its Krull dimension. Every regular local ring is an integral domain and a unique factorization domain.

In a regular local ring of dimension $d$, a regular system of parameters is a list in the maximal ideal whose images form a residue-field basis of $\mathfrak{m}/{\mathfrak{m}}^2$. The Nakayama lemma shows that it generates the maximal ideal. At a smooth closed point over an algebraically closed field, these are the local parameters in all dimensions, extending the single local parameter on a smooth algebraic curve.

A chosen regular system of parameters gives the displayed completion isomorphism by sending $T_i$ to $t_i$. The associated graded ring of a regular local ring is the polynomial ring on their initial forms. Successively removing the homogeneous initial form of each remainder gives a unique formal series for every local function. The Krull intersection theorem embeds the local ring in its completion. The power-series characterization is also given in Stacks Project Lemma 10.160.10.

Let $(A,\mathfrak{m})$ be a regular local ring of Krull dimension $d$. Its formal power series ring $B=A[[x]]$ is Noetherian and local, with maximal ideal $I=\mathfrak{m}B+(x)$ and embedding dimension $d+1$. Since $I$ is generated by $d+1$ elements, the Krull height theorem gives $\dim B\le d+1$. For the reverse inequality choose a strict prime chain ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_d=\mathfrak{m}$ in $A$. Finite generation of each ${\mathfrak{p}}_i$ gives $B/{\mathfrak{p}}_{i}B\cong (A/{\mathfrak{p}}_i)[[x]]$, an integral domain, so each extended ideal is prime. The inclusions remain strict by contraction to $A$, and $\mathfrak{m}B⊊I$ adds one more step since $x\notin \mathfrak{m}B$. This proves $\dim B=d+1$, equal to its embedding dimension, and hence regularity.

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.