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Gabrielov’s rank condition is equivalent to an inequality of reduced orders. (English) Zbl 0612.32013

Let \(\Phi: (Y,\eta)\to (X,\xi)\) be a morphism of germs of complex spaces. The paper under review relates the order of vanishing of the germ of holomorphic function f on x at \(\xi\), to the order of vanishing of the germ \(f\circ \Phi\) on Y at \(\eta\). Let \(\phi\) denote the naturally induced homorphism on the function rings \((\phi (f)=f\circ \Phi)\), and suppose \(\phi\) is injective. The author derives necessary and sufficient conditions for the existence of a number \(n\geq 1\) such that \({\bar \nu}\)(\(\phi\) (f))\(\leq n{\bar \nu}(f)\) for arbitrary f. Here \({\bar \nu}\) is the reduced order. The author defines the componentwise geometric rank of \(\phi\), and shows such n exists if and only if this rank is full.
Reviewer: G.Harris

MSC:

32A38 Algebras of holomorphic functions of several complex variables
13J05 Power series rings
13H05 Regular local rings
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References:

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