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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On Noetherianness of Nash rings


Authors: Fulvio Mora and Mario Raimondo
Journal: Proc. Amer. Math. Soc. 90 (1984), 30-34
MSC: Primary 13E05; Secondary 14G30, 32B05, 58A07
DOI: https://doi.org/10.1090/S0002-9939-1984-0722409-X
MathSciNet review: 722409
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0722409-X
Keywords: Nash rings, noetherian property, algebraic closure of rings
Article copyright: © Copyright 1984 American Mathematical Society