## Analytic rings.(English)Zbl 0575.32004

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

Zbl 0177.179
Full Text:

### References:

 [1] 1 Alfhors , L.V. , Complex Analysis , Mc Graw Hill , 1966 . [2] 2 Cartan , H. , Teoria Elemental de las Funciones Analiticas a una y varias variables complejas , Selecciones Cinetificas , 1968 . [3] 3 Dubuc , E. , Sur les modèles de la Géométrie Différentielle Synthétique , Cahiers Top. et Géom. Diff. XX- 3 ( 1979 ). Numdam | MR 557083 | Zbl 0473.18008 · Zbl 0473.18008 [4] 4 Gabriel , P. & Zisman , M. , Calculus of fractions and Homotopy theory , Springer , 1976 . Zbl 0186.56802 · Zbl 0186.56802 [5] 5 Grauert , H. & Fritsche , K. , Several complex variables , Springer , 1976 . MR 414912 | Zbl 0381.32001 · Zbl 0381.32001 [6] 6 Gunning , R.C. & Rossi , H. , Analytic functions of several complex variables , Prentice Hall , 1965 . MR 180696 | Zbl 0141.08601 · Zbl 0141.08601 [7] 7 Malgrange , B. , Ideals of differentiable functions , Oxford University Press , 1966 . MR 212575 | Zbl 0177.17902 · Zbl 0177.17902 [8] 8 Malgrange , B. , Analytic spaces, Monographie 17 de l ’ Enseignement Math. , Genève , 1968 . MR 237824 [9] 9 Weil , A. , Théorie des points proches sur les variétés différentiables , Colloque Top. et Géom. Diff. Strasbourg , ( 1953 ), 111 - 117 . MR 61455 | Zbl 0053.24903 · Zbl 0053.24903
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