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