Analytic rings. (English) Zbl 0575.32004

Dubuc, Eduardo; Taubin, Gabriel

Cah. Topol. Géom. Différ. 24, 225-265 (1983).

An analytic ring in a category \({\mathcal E}\) is defined as a functor \({\mathcal C}\to {\mathcal E}\), defined over the category \({\mathcal C}\) of open subsets of \({\mathbb{C}}^ n\), \(n\in N\), and holomorphic functions. This functor is required to preserve all transversal pullbacks that may exist in \({\mathcal C}.\)
Examples of analytic rings are the following: rings of holomorphic functions on an open set of \({\mathbb{C}}^ n\), or more generally, on any complex manifold. Local rings of germs of holomorphic functions, as well as analytic algebras in the sense of B. Malgrange [”Ideals of differentiable functions”, (1966; Zbl 0177.179)]. In particular, all complex Weil algebras. Rings of sections of any analytic space. The sheaf of continuous complex valued functions on any topological space X is an analytic ring in the topos of sheaves over X. As well as the structure sheaf of any analytic space. Also, the inclusion from \({\mathcal C}\) into the category of analytic spaces is an analytic ring in this category.
In the first part of the paper the authors give the definition of analytic rings and some of their basic properties, and they treat the local analytic rings. In the second they consider the analytic spaces. Finally, in the third, they construct the generic analytic ring in a category with finite limits. That is, the (algebraic) theory of analysis rings.

MSC:

32B05	Analytic algebras and generalizations, preparation theorems
46E25	Rings and algebras of continuous, differentiable or analytic functions
32A38	Algebras of holomorphic functions of several complex variables
18F15	Abstract manifolds and fiber bundles (category-theoretic aspects)
18F20	Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
30H05	Spaces of bounded analytic functions of one complex variable

Keywords:

transversal pullbacks; analytic rings

Citations:

Zbl 0177.179

References:

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