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The rank condition and convergence of formal functions. (English) Zbl 0688.32008

From the author’s introduction: “In this paper we treat two facts concerning convergence of formal holomorphic functions. One is that, for a germ of holomorphic mappings with maximal generic rank, the formal function at the target converges if its pullback does so (Gabrielov). The other is propagation of convergence along an exceptional set.
Let \(\Phi\) : \(Y\to X\) be a morphism between complex spaces and \(\phi\) : A\(={\mathcal O}_{x,\xi}\to B={\mathcal O}_{y,\eta}\) the induced homomorphism between analytic local algebras. To simplify the problem, assume that the germs \(X_{\xi}\) and \(Y_{\eta}\) are reduced and irreducible, i.e., A and B are integral domains. The author has been interested in equivalence of the following conditions:
(0) \(\exists a\geq 1\), \(\exists b\geq 0\), \(\forall f\in A:\) \(a\nu (f)+b\geq \nu (\phi (f))\), where \(\nu\) (f) denotes the vanishing order of f at the point.
(1) Gabrielov’s strong rank condition: The image \(\Phi (Y_{\eta})\subset X_{\xi}\) has full topological dimension, i.e., grk \(\phi\) \(=\dim X_{\xi}.\)
(2) \({\hat \phi}{}^{-1}(B)=A\), i.e., \(f\in \hat A\) is convergent if its pullback is convergent, where \({\hat \phi}\): \(\hat A\to \hat B\) denotes the canonical extension to the completions.
(3) \(\phi\) is a closed embedding with respect to the Krull topology.
(4) \(\phi\) is injective and \(\phi\) (A) is closed in B.”
The following main results are proved: Theorem (9.2). (1) \(\Leftrightarrow\) (2) (\(\Leftrightarrow\) (3)). Theorem (9.5). If \(f\in H^ 0(E,\hat {\mathcal O})\) is convergent at some \(\xi\), it is convergent everywhere on E, i.e., \(f\in H^ 0(E,{\mathcal O})\).
Reviewer: M.S.Marinov

MSC:

32B10 Germs of analytic sets, local parametrization
32B05 Analytic algebras and generalizations, preparation theorems
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