On Noetherianness of Nash rings

Abstract:

We introduce a class of rings, called Nash Rings, which generalize the notation of rings of Nash functions. Let $k$ be any field, $X$ be a normal algebraic variety in ${k^n}$, and $U \subset X$. A Nash ring $D$ is the algebraic closure of $\Gamma (X,{\mathcal {O}_X})$ in a suitable domain $B$ such that $U$ is contained in the maximal spectrum of $B$ and $\Gamma (X,{\mathcal {O}_X})$ is analytically isomorphic to $B$ at each $x \in U$. We show that $D$ is a ring of fractions of the integral closure of $\Gamma (X,{\mathcal {O}_X})$ in $B$. Moreover, if $k$ is algebraically nonclosed and if every algebraic subvariety $V \subset X$ intersects $U$ in a finite number of connected components (in the topology induced by $B$), then $D$ is noetherian.