## 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
Full Text:

### References:

 [1] Eakin, P.M., Harris, G.A.: When ?(f) convergent impliesf is convergent. Math. Ann.229, 201-210 (1977) · Zbl 0355.13010 [2] Gabrielov, A.M.: Formal relations between analytic functions. Izv. Akad. Nauk. SSSR37, 1056-1088 (1973) · Zbl 0297.32007 [3] Hironaka, H.: Resolutions of singularities of an algebraic variety over a field of characteristic zero. Ann. Math.79, 109-326 (1964) · Zbl 0122.38603 [4] Izumi, S.: Linear complementary inequalities for orders of germs of analytic functions. Invent. Math.65, 459-471 (1982) · Zbl 0497.32005 [5] Izumi, S.: A measure of integrity for local analytic algebras. Publ. RIMS, Kyoto Univ.21, 719-735 (1985) · Zbl 0587.32016 [6] Lejeune-Jalabert, M., Teissier, B.: Cl?ture int?grale des id?aux et ?quisingularit? (S?m. au Centre de Math. ?cole Polytechnique, 1974). Pub. du Laboratoire de math. pures de l’Universit? Scientifique et M?dicale de Grenoble [7] Nagata, M.: Note on a paper of Samuel concerning asymptotic properties of ideals. Mem. Coll. Sci. Univ. Kyoto, Ser. A30, 165-175 (1957) · Zbl 0095.02305 [8] Rees, D.: Valuations associated with a local ring. I. Proc. London Math. Soc.5, 107-128 (1955) · Zbl 0066.28805 [9] Rees, D.: Valuations associated with a local ring. II. J. London Math. Soc.31, 228-235 (1956) · Zbl 0074.26401 [10] Rees, D.: Izumi’s theorem. Unpublished (1985) · Zbl 0741.13011 [11] Risler, J-J.: Les exposants de ?ojasiewitz dans le cas analytique r?el (Appendix of [6])
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.