Open embeddings of Stein spaces and of C∞-manifolds are exactly the maps whose induced homomorphism of function algebras is a 1-pseudoflat epimorphism, with equivalent homological conditions in the smooth case.
Flat ring epimorphisms and universal localisations of commutative rings
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abstract
We study different types of localisations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localisation in the sense of Cohn and Schofield; and (b) when such universal localisations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialisation closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localisations. Moreover, it turns out that an answer to the question of when universal localisations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localisation. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.
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Open embeddings and pseudoflat epimorphisms
Open embeddings of Stein spaces and of C∞-manifolds are exactly the maps whose induced homomorphism of function algebras is a 1-pseudoflat epimorphism, with equivalent homological conditions in the smooth case.