RT Journal Article
T1 Separation, factorization and finite sheaves on Nash manifolds
A1 Coste, M.
A1 Ruiz Sancho, Jesús María
A1 Shiota, Masahiro
AB 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.
PB Cambridge University Press
SN 0010-437X
YR 1996
FD 1996-08
LK https://hdl.handle.net/20.500.14352/58304
UL https://hdl.handle.net/20.500.14352/58304
LA eng
NO DGICYT
DS Docta Complutense
RD 8 abr 2025