Separation, factorization and finite sheaves on Nash manifolds

Abstract

Nash functions are those real analytic functions which are algebraic over the polynomials. Let M⊂Rn be a Nash manifold, N(M) the ring of Nash functions on M and O(M) the ring of analytic functions on M. The following problems have been open for at least twenty years: (1) Separation problem: Let G be a prime ideal of N(M); is G⋅O(M) a prime ideal? (2) Factorisation problem: Given f∈N(M) and an analytic factorisation f=f1⋅f2, do there exist Nash functions g1,g2 on M and positive analytic functions φ1,φ2 such that φ1⋅φ2=1 and f1=φ1g1, f2=φ2g2? (3) Global equations problem: Is every finite sheaf I of ideals of N generated by global Nash functions? (4) Extension problem: For the same I as above, is the natural homomorphism H0(M,N)→H0(M,N/I) surjective? The main results of this paper are: Theorem. For any Nash manifold M, Problem 1 has a positive answer if and only if Problem 3 (or Problem 4) have a positive answer. Problem 3 has a positive answer for any locally principal finite sheaf if and only if Problem 2 has a positive answer. It is interesting to remark that the authors, in a recent paper, have proved that the above problems have a positive answer under the hypothesis that M be compact.