## Algebraic geometry and Nash functions |

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abelian groups absolutely reduced affine open affine representation affine scheme affine subvariety affine variety defined afflne algebraically closed field analytic functions analytic set analytic variety called category of A-coherent closed sets closed subset coherent sheaf cohomology theory compact completion complexification Corollary defined over Q denned denote determines e-approximation equation equivalent exact sequence extension finite type finitely presented following result germ Vx Hence homomorphism ideal of definition imbedding irreducible isomorphic K-algebra K[Xl K[Xl,Xn K[Xx Let 3r maximal ideal module morphism Nullstellensatz open covering open set open sets XP open subset Ox-modules polynomial prescheme presheaf prime ideal Proof proposition proves the lemma quasi-regular real analytic space regular affine variety regular function Remark representation of rv ringed space rv(Ov rx(Ox sheaf of ideals Spec rv Step surjective tion topological space V Q K XN+l Zariski topology ZC(P