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Local analytic rings. (English) Zbl 0705.32002

The author develops some aspects of the theory of analytic rings in the sense of E. J. Dubuc and G. Taubin [Cah. Topologie Géom. Différ. 24, 225-265 (1983; Zbl 0575.32004)]. If C is the category of open subsets of \({\mathbb{C}}^ n\) (with holomorphic functions as morphisms) and E is a category, an analytic ring in E is a functor which preserves transversal pullbacks and terminal objects (or, equivalently, preserves independent equalizers, finite products and open inclusions). If A is an analytic ring in E, it is called local iff for each open covering \((U_{\alpha})_{\alpha \in I}\) of an open set \(U\subset {\mathbb{C}}^ n\), the family \((A(U_{\alpha})\to A(U))_{\alpha \in J}\) is a universal effective epimorphic family in E. This definition is equivalent to the one considered in the above quoted paper.
The author explicitely constructs the classifying topos of the theory of local analytic rings. Further results, examples and open problems are presented.
Reviewer: E.Pascu

MSC:

32B05 Analytic algebras and generalizations, preparation theorems
18B25 Topoi
18F05 Local categories and functors

Citations:

Zbl 0575.32004
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References:

[1] 1 Artin , M. , Grothendieck , A. & Verdier , J.L. , Théorie des topos et cohomologie étale des schèmes (S.G.A. 4) , Lecture Notes in Math. 269 , Springer ( 1972 ). MR 354653
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[3] 3 Dubuc , E. & Taubin , G. , Analytic rings , Cahiers Top. et Géom. Diff. XXIV - 3 ( 1983 ). Numdam | MR 728632 | Zbl 0575.32004 · Zbl 0575.32004
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