Regular system of parameters ($(t_1,\dots ,t_d)$)

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.