Flatness in analytic mappings. I: On an announcement of Hironaka
type: Item
instance of: scholarly article
title: Flatness in analytic mappings. I: On an announcement of Hironaka
zbMATH Open document ID: 0728.32007
DOI: 10.1007/BF02938113
author: Edward Bierstone
author: Pierre Milman
published in: The Journal of Geometric Analysis
publication date: 1991-01-01T00:00:00Z
review text: In the research announcement in Proc. Japan Acad., Ser. A 62, 73-75 (1986; Zbl 0589.32015), \textit{H. Hironaka} asserts the existence of a ``semicoherent'' stratification for a subanalytic subset. The sketch of the proof contains three steps. In the paper under review the authors prove that the accouncement of Hironaka is not correct. In fact they prove the first step, namely: Theorem. Let \(\Phi: Y\to M\) be a proper morphism of real analytic spaces with M smooth and \(X=\Phi (Y)\) (which is closed and subanalytic). Let \({\mathcal F}\) be a coherent sheaf of \({\mathcal O}_ Y\)-modules. Then there exists a locally finite stratification of X, say \(X=\cup_{i}X_ i\), such that for each \(T=X_ i:\) (1) T is a locally closed real analytic submanifold of M which is connected and subanalytic in M. (2) Let \({\mathcal H}_ T\) denote the ideal sheaf in \({\mathcal O}_ Y\) (within a small neighborhood of \(\Phi^{-1}(T))\) generated by the ideal sheaf of T in \({\mathcal O}_ M\) via \(\Phi\). Then the restriction to \(\Phi^{-1}(T)\) of \(\oplus^{\infty}_{k=0}{\mathcal H}^ k_ T\cdot {\mathcal F}/{\mathcal H}_ T^{k+1}\cdot {\mathcal F}\) is flat as a \({\mathcal O}_ T\)-module. However, by counterexamples, it is shown that both step 2 and 3 in Hironaka's announcement are not correct. The proof of the theorem above is based on a generic normal flatness result which is proved using semicontinuity properties of the ``diagram of initial exponents'' and also properties of the parametrized families formal of power series (earlier developed by the authors).
reviewed by: Nicolae Mihalache
Mathematics Subject Classification ID: 32B20
Mathematics Subject Classification ID: 14B25
zbMATH DE Number: 4202692
zbMATH Keywords: subanalytic set
zbMATH Keywords: stratification
zbMATH Keywords: flatness
MaRDI profile type: Publication
cites work: Relations among analytic functions. I
cites work: Ideals of Holomorphic Functions with C ∞ Boundary Values on a Pseudoconvex Domain
cites work: Semianalytic and subanalytic sets
cites work: Q4405003
cites work: Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné
cites work: Points de platitude d'un morphisme d'espaces analytiques complexes
cites work: Formal relations between analytic functions
cites work: FORMAL RELATIONS BETWEEN ANALYTIC FUNCTIONS
cites work: Q5638554
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Q4142041
cites work: Local analytic dimensions of a subanalytic set
cites work: Q4014102
description: scientific article; zbMATH DE number 4202692