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The global properties are more difficult to obtain. The fact that the ring of Nash functions on a Nash manifold (even non compact) is [[noetherian ring|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 manifold]]s. Let <math>\mathcal{N}</math> denote the sheaf of Nash function germs on
a Nash manifold ''M'', and <math>\mathcal{I}</math> be a [[coherent sheaf]] of <math>\mathcal{N}</math>-ideals. Assume <math>\mathcal{I}</math> is finite, i.e., there exists a finite open semialgebraic covering <math>\{U_i\}</math> of ''M'' such that, for each ''i'', <math>\mathcal{I}|_{U_i}</math> is generated by Nash functions on <math>U_i</math>. Then <math>\mathcal{I}</math> is globally generated by Nash functions on ''M'', and the natural map
:::<math>H^0(M,\mathcal{N}) \to H^0(M,\mathcal{N}/\mathcal{I})</math>
is surjective. However
:::<math>H^1(M,\mathcal{N})\neq 0, \ \text{if} \ \dim(M) > 0,</math>
contrarily to the case of Stein manifolds.
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