Flatness criteria
type: Item
instance of: scholarly article
title: Flatness criteria
zbMATH Open document ID: 0853.14008
author: János Kollár
published in: Journal of Algebra
publication date: 1997-01-06T00:00:00Z
review text: Let \(f : (X,x) \to (Y,y)\) be a morphism of local schemes and \((Z,x)\) the central fiber. The flatness of \(f\) is a subtle geometric notion and it is difficult to find some flatness criteria which do not require infinitesimal properties. A well-known result of \textit{H. Hironaka} [Ill. J. Math. 2, 355-366 (1958; Zbl 0099.15701)] gives such a criterion: Assume that (i) \(Z\) is generically reduced and \(\text{red} Z\) is normal; (ii) \(Y\) is regular and \(X\) has no embedded points, and (iii) \(f\) is of pure dimension; then \(f\) is flat and \(Z\) is normal. But this criterion is not always sufficient in the study of the deformations of canonical varieties or when we compare the Hilbert schemes and the Chow varieties [cf. \textit{J. Kollár}, ``Rational curves on algebraic varieties'', Springer, Berlin (1995)]. In the paper under review, the author gives some generalized criteria of flatness. For example, when \(Y\) is excellent, we can assume in the Hironaka criteria that \(Y\) is normal (instead of regular) and, if \(\text{char} k(y) > 0\), that \(f\) is essentially of finite type, with the same conclusion: \(f\) is flat and \(Z\) normal. Some versions with sheaves are also given.
Mathematics Subject Classification ID: 14E99
zbMATH DE Number: 783944
zbMATH Keywords: deformations
zbMATH Keywords: flatness of morphism
zbMATH Keywords: Hilbert schemes
zbMATH Keywords: Chow varieties
reviewed by: J.-Y. Mérindol
MaRDI profile type: Publication
full work available at URL: jabr.1995.1209
OpenAlex ID: W4212900573
DOI: 10.1006/JABR.1995.1209
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description: scientific article; zbMATH DE number 783944