Global Properties
The global properties are more difficult to obtain. The fact that the ring of Nash functions on a Nash manifold (even noncompact) is noetherian was proved independently (1973) by Jean-Jacques Risler and Gustave Efroymson. Nash manifolds have properties similar to but weaker than Cartan's theorems A and B on Stein manifolds. Let denote the sheaf of Nash function germs on a Nash manifold M, and be a coherent sheaf of -ideals. Assume is finite, i.e., there exists a finite open semialgebraic covering of M such that, for each i, is generated by Nash functions on . Then is globally generated by Nash functions on M, and the natural map
is surjective. However
contrarily to the case of Stein manifolds.
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