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Proper morphisms and excellent schemes

Published online by Cambridge University Press:  22 January 2016

Barbara Bellaccini
Barbara Bellaccini*
Affiliation:
Istituto Matematico Università di Siena Italy
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Let f: X → Y be a finite type morphism of locally noetherian schemes. It is well known ([3, IV, 7.8.6]) that the excellent property ascends from Y to X. On the other side there are counter-examples where X is excellent and Y is not. First of all it is easy to show that the condition on chains of prime ideals does not descend (see [3, IV, 7.8.4]), even by finite morphisms. Secondly in [2] it is produced an example where X is excellent while Y is not a G-scheme (i.e. it has not the good properties of formal fibers). However in [2] it is also proved that the property concerning the openness of regular loci (the so called “J-2”) descends by finite type surjective morphisms. Therefore we are led to the following question: When does the G-scheme property descend? I.e. what conditions do we need on f? A reasonable condition is conjectured (in [2]) as the following: f is proper surjective. The aim of the present paper is precisely to give an answer to such a question. What we really prove is the following. If X is a G-scheme and J-2 (quasi-excellent), then the same is true for Y, provided that f is proper surjective and moreover all the residue fields of Y have characteristic 0. We remark that the result is strongly based on Hironaka’s desingularization for quasi-excellent schemes defined over a field of characteristic 0.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 89 , March 1983 , pp. 109 - 118
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

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