# Analytic ring

A ring of germs of analytic functions at a point in an analytic space. The following is a more accurate definition. Let $k$ be a field with a non-trivial norm (cf. Norm on a field) (which is usually assumed to be complete), and let $k\{\{X_1,\dots,X_n\}\}$ be the $k$-algebra of power series in $X_1,\dots,X_n$ with coefficients in $k$ which converge on some polycylinder with centre $(0,\dots,0)$, each series converging on its own polycylinder. A quotient ring of the ring $k\{\{X_1,\dots,X_n\}\}$ is called an analytic ring over $k$, or an analytic $k$-algebra; usually, $k$ is the field of real numbers $\mathbf R$ or the field of complex numbers $\mathbf C$. Any analytic ring is a local, Noetherian, Hensel ring; its field of residues is isomorphic to $k$. An analytic ring $k\{\{X_1,\dots,X_n\}\}$ is a regular (and a factorial) ring, and its completion in the topology defined by the maximal ideal $(X_1,\dots,X_n)$ coincides with the ring of formal power series $k[[X_1,\dots,X_n]]$. The normalization lemma is true: An integral analytic ring is a finite extension of an analytic ring $k\{\{X_1,\dots,X_n\}\}$. Algebras that are finite over $k\{\{X_1,\dots,X_n\}\}$ are generally called quasi-analytic $k$-algebras. If $k$ is a perfect field, an analytic ring is an excellent ring.