Equimultiplicity and Blowing up An Algebraic Study

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2 M. Herrmann S. Ikeda U. Orbanz Equimultiplicity and Blowing up An Algebraic Study With an Appendix by B. Moonen Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

3 Manfred Herrmann Ulrich Orbanz Mathematisches Institut der UniversitiH zu K61n Weyertal 86-90, K61n 41, FRG Shin Ikeda Mathematical Department Gifu College of Education 2078 Takakuwa, Gifu, Japan With 11 Figures The figure on the cover illustrates Theorem (20.5) of Chapter IV. The geometry of this is elaborated in Chapter 111,2.2 of the Appendix, see in particular Theorem (2.2.2) and (2.2.32). Mathematics Subject Classification (1980): 13H10, 13H15, 14805,14815,32805,32830 IS8N-13: e-is8n-13: : / Libary of Congress Cataloging-in-Publication Data. Herrmann, Manfred, Equimultiplicity and blowing up. Bibliography: p. Includes index. 1. Multiplicity (Mathematics) 2. Blowing up (Algebraic geometry) 3. Local rings. I. Ikeda, S. (Shin), Orbanz, Ulrich, Title. QA H This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its version of June 24,1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. Springer-Verlag Berlin Heidelberg 1988 Softcover reprint of the hardcover 1 st edition /

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5 Preface This book is intended as a special course in commutative algebra and assumes only a general familiarity with topics on commutative algebra and algebraic geometry included in textbooks. We treat two kinds of problems. One of them consists in controlling Hilbert functions after blowing up convenient centers. This question arises directly from the resolution of algebraic and complex-analytic singularities. The other problem is to investigate Cohen-Macaulay properties under blowing up. We begin with some remarks on the background. 1) In the case of plane curves desingularization means classification of singularities, since by blowing up points we get a multiplicity sequence which describes the topological type of the singularity. For the case of higher dimensions and codimensions Zariski and Hironaka suggested to blow up regular centers D contained in the singular locus of the given variety X In this case the hierarchy of numerical conditions on D is as follows: (i) all points of D have the same multiplicity (ii) all points of D have the same Hilbert polynomial (iii) all points of D have the same Hilbert functions. These three conditions coincide for hypersurfaces but they differ in general. For each condition there is an algebraic description, namely by reductions of ideals for (i) and by flatness conditions on the associated graded ring for (ii) and (iii) Hironaka's inductive resolution of algebraic schemes over fields of characteristic zero makes use of numerical conditions arising from the Hilbert functions. The approach to the problem by a non-inductive procedure is open and one is still far from the knowledge of complexity and classification of singularities of dimension bigger than one. Besides blowing up regular centers there are also approaches to desingularization which amount to blowing up singular centers; for example: a) Zariski-J~ng's desingularization of surfaces in characteristic zero, using generic projections and embedded resolution of the diseriminant locus, induces blowing ups at singular centers on the surface.

6 VIII b) Also in the way of desingularization by blowing up non-regular centers one can mention M. Spiva!:ovsky' s resolution of two-dimensional complex-analytic singularities by Nash transformations and normalizations. In order to control singularities under blowing up singular centers one can ask for generalizations of the numerical conditions (i), (ii) or (iii) and their algebraic descriptions. That we do by using generalized Hilbert functions and multiplicities. This allows to extend many classical results to a more general situation, and it leads us to three essential types of numerical conditions as three possibilities to make precise the naive idea of "equimultiplicity". Note that the role of multiplicities and Hilbert functions in geometry is that they furnish some way of measuring and comparing singularities. The concept of multiplicity is older than that of the Hilbert function, but since Samuel has shown how to compute multiplicities via Hilbert functions, many results on multiplicities are consequences of the corresponding results for Hilbert functions. Still there are some results on multiplicities which are not parallel to Hilbert functions, due to the following facts: a) The multiplicity of a local ring is always the degree of a generic projection, which means algebraically that there is a system of parameters giving the same multiplicity as the maximal ideal. b) For multiplicities there is a projection formula for finite morphisms. (There is nothing similar to a) and b) for Hilbert functions, of course.) c) To derive a relation between multiplicities from Hilbert functions, one needs to know something on the dimensions, which occur as degrees of the Hilbert polynomials. In particular, lower dimensional components do not enter into the multiplicity. Therefore already Chevalley assumed his local rings to be quasi unmixed. In numerous papers, Ratliff has developped a fairly complete theory for quasi-unmixed rings, and it is now clear that the notion of quasi-unmixedness gives the correct frame for the study of multiplicities. 2) Let X' be a blowing up of a variety X with center Y. In general the Cohen-Macaulay properties of X and X' are totally unrelated. But if Y is locally a complete intersection and if the local cohomology modules of the affine vertex over X (or the local cohomology of the vertex of the conormal cone of Yare finitely generated in all orders :;; dim X (or < dim X respectively) then X I may become Cohen-Macaulay. This gives a motivation to study arithmetic properties as well as geometric ones of X' and its exceptional divisor

7 IX The main purpose of the Appendix by B. Moonen is to provide a geometric description of the notion of multiplicity and a geometric interpretation of the notion of an equimultiple ideal within the realm of complex analytic geometry. Now we give a detailed description of the contents of the book. Chapter I - III contain the basic techniques except local duality which is treated in Chapter VII. In Chapter I we recall all the basic facts about multiplicities, Hilbert functions and reductions of ideals. The second Chapter contains some general facts about graded rings that arise in connection with blowing up. We also recall the theory of standard bases. In Chapter III several characterizations of quasiunmixed local rings are given. A very useful tool for these characterizations is the theory of asymptotic sequences which has been mainly developed by Ratliff and Katz. Our treatment follows closely the way of Katz. Chapter IV presents various notions of equimultiplicity. For a hyper surface and a regular subvariety there exists a "natural" notion of equimultiplicity, and there are different directions of generalization: a) to the non-hyper surface case, b) to non-regular subvarieties. In these more general situations there are weaker and stronger notions, all of which specialize to equimultiplicity in the original case. We mention three essentially different algebraic generalizations of equimultiplicity together with a numerical description of each condition. Then we describe the hierarchy among these conditions. Finally we investigate these conditions concerning openess and transitivity properties. Chapter V shows that these conditions are of some use to investigate Cohen-Macaulay properties under blowing up. In Chapter VI we indicate that the new conditions of equimultiplicity are useful in the study of the numerical behaviour of singularities under blowing up singular centers. In this context we consider two essential topics: blowing up and semicontinuity. To prove semicontinuity one has to desingularize curves by blowing up points, and conversely for inequalities of Hilbert functions under blowing up other centers one has to use semicontinuity. Chapter VII, presupposing the following Chapters VIII and IX, discusses local cohomology and duality over graded rings. For local rings, the theory of local duality can be found in textbooks. For the corresponding results over graded rings we give detailed proofs because they

8 x are not all out available in the literature. Chapter VIII studies local rings (A,m) with finite local cohomology Hi (A) for i" dim A If X is an irreducible non-singular projective variety over a field then the local ring at the vertex of the affine cone over X has always this property. We present the results on these rings in a unified manner according to S. Goto and N.V. Trung. In Chapter IX the results of Chapter V are partially extended and rephrased in a different context by using cohomological methods. The main result is a general criterion of the Cohen-Macaulay property of Rees rings. Then we investigate Rees rings of certain equimultiple ideals. Finally we give special applications to rings with low multiplicities. In this context we also prove the equivalence between the "monomial property" and the "direct summand-property" in the sense of M. Hochster. The Appendix consists of three parts. Part I treats the fundamentals of local complex analytic geometry in a fairly complete way, for the sake of reference, and convenience for the reader. Some emphasis is laid on effective methods, and so consequent use of the general Division Theorem, due to Grauert-Hironaka, is made. Part II exposes the geometric description of the multiplicity of a complex spacegerm as the local mapping degree of a generic projection. To handle the nonreduced case, the notion of compact Stein neighbourhoods is introduced, which allow a systematic transition from the algebraic to the analytic case. The connection with Samuel multiplicity is described. Part III develops the theory of compact Stein neighbourhoods further and thus deduces the properties of normal flatness in the analytic case from the algebraic case. Then the geometry of equimultiplicity along a smooth subspace is developed in some detail in 2 with geometric proofs. Finally, 3 treats the geometric content of the equimultplicity results of Chapter IV; these follow from the algebraic results via the method of compact Stein neighbourhoods. We have to make some acknowledgments. First we would like to express our gratitude in particular to J. Giraud, J. Lipman, R. Sharp and J.L. Vicente for suggestions and encouragements during the preparation of this work. Furthermore we thank deeply D. Katz, L. Robbiano, O. Villamayor and K. Yamagishi for their careful reading of various positions of the manuscript or for their detailed suggestions and improvements. K. Yamagishi also worked out the main part

9 of the appendix to Chapter V. Finally we have to emphasize the help of our students F. Bienefeldt, D. Rogoss, M. Ribbe and M. Zacher. Their stimulating questions in the seminars and their special contributions to the Chapters VII, VIII and IX (besides reading carefully the manuscript) have essentially improved the first version of the last three chapters. Chapter V contains the main results of the thesis of Dr. U. Grothe who worked out the main part of this chapter. Last not least we owe thanks to Mrs. Pearce from the Max-Planck-Institute of Mathematics in Bonn for typing services and for patience and skill. The third author has received support and great hospitality by the Max-Planck-Institute of Mathematics in Bonn by the Department of Mathematics of the University in Genova and the Department of Mathematics of the University of Kansas. The acknowledgements of the author of the Appendix are stated in the introduction there.

10 Table of contents Chapter I REVIEW OF MULTIPLICITY THEORY.... The multiplicity symbol... 1 Hilbert functions Generalized reultiplicities and Hilbert functions Reductions and integral closure of ideals Faithfully flat extensions Projection formula and criterion for multiplicity one Examples Chapter II - Z-GRADED RINGS AND MODULES Associated graded rings and Rees algebras Dimension Homogeneous parameters Regular sequences on graded modules Review on blowing up Standard bases Examples 100 Appendix - Homogeneous subrings of a homogeneous ring Chapter III - ASYMPTOTIC SEQUENCES AND QUASI-UNMIXED RINGS Auxiliary results on integral dependence of ideals Associated primes of the integral closure of powers of an ideal..." Asymptotic sequences Quasi-unmixed rings The theorem of Rees-Boger Chapter IV - VARIOUS NOTIONS OF EQUIMULTIPLE AND PERMISSIBLE IDEALS Reinterpretation of the theorem of Rees-Boger Hironaka-Grothendieck homomorphism Projective normal flatness and numerical characterization of permissibility Hierarchy of equimultiplicity and permissibility Open conditions and transitivity properties

11 XIV Chapter V - EQUIMULTIPLICITY AND COHEN-MACAULAY PROPERTY OF BLOWING UP RINGS Graded Cohen-Macaulay rings The case of hypersurfaces Transitivity of Cohen-Macaulayness of Rees rings Appendix (K. Yamagishi and U. Orbanz) - Homogeneous domains of minimal multiplicity Chapter VI - CERTAIN INEQUALITIES AND EQUALITIES OF HILBERT FUNCTIONS AND MULTIPLICITIES Hyperplane sections Quadratic transformations Semicontinuity Permissibility and blowing up of ideals Transversal ideals and flat families Chapter VII - LOCAL COHOMOLOGY AND DUALITY OF GRADED RINGS Review on graded modules Matlis duality Part I Local case Part II: Graded case Local cohomology Local duality for graded rings Appendix - Characterization of local Gorenstein-rings by its injective dimension 320 Chapter VIII - GENERALIZED COHEN-MACAULAY RINGS AND BLOWING UP Finiteness of local cohomology Standard system of parameters The computation of local cohomology of generalized Cohen-Macaulay rings Blowing up of a standard system of parameters Standard ideals on Buchsbaum rings Examples

12 xv Chapter IX - APPLICATIONS OF LOCAL COHOMOLOGY TO THE COHEN-MACAULAY BEHAVIOUR OF BLOWING UP RINGS Generalized Cohen-Macaulay rings with respect to an ideal The Cohen-Macaulay property of Rees algebras Rees algebras of m-primary ideals The Rees algebra of parameter ideals The Rees algebra of powers of parameter ideals Applications to rings of low multiplicity Examples Appendix (B. Moonen) - GEOMETRIC EQUIMULTIPLICITY INTRODUCTION 448 I. LOCAL COMPLEX ANALYTIC GEOMETRy Local analytic algebras Formal power series Convergent power series Local analytic Jk-algebras Local WeierstraB Theory I: The Division Theorem Ordering the monomials Monomial ideals and leitideals The Division Theorem Division with respect to an ideal; standard bases Applications of standard bases: the General WeierstraB Preparation Theorem and the Krull Intersection Theorem The classical WeierstraB Theorems Complex spaces and the Equivalence Theorem Complex spaces Constructions in c:;el The Equivalence Theorem The analytic spectrum Local WeierstraB Theory II: Finite morphisms Finite morphisms WeierstraB maps The Finite Mapping Theorem The Integrality Theorem

13 XVI 5. Dimension and Nullstellensatz Local dimension Active elements and the Active Lero~a The Ruckert Nullstellensatz Analytic sets and local decomposition The Local Representation Theorem for complex space-germs (Noether normalization) Openness and dimension Geometric interpretation of the local dimension and of a system of parameters; algebraic Noether normalization The Local Representation Theorem; geometric Noether normalization Coherence Coherent sheaves Nonzerodivisors Purity of dimension and local decomposition Reduction II. GEOMETRIC MULTIPLICITY Compact Stein neighbourhoods Coherent sheaves on closed subsets Stein subsets Compact Stein subsets and the Flatness Theorem Existence of compact Stein neighbourhoods Local mapping degree Local decomposition revisited Local mapping degree Geometric multiplicity The tangent cone Multiplicity The geometry of Samuel multiplicity Degree of a projective variety Hilbert functions A generalization Samuel multiplicity Algebraic multiplicity Algebraic degree Algebraic multiplicity

14 XVII III. GEOMETRIC EQUIMULTIPLICITY Normal flatness and pseudoflatness Generalities from Complex Analytic Geometry The analytic and projective analytic spectrum Flatness of admissible graded algebras The normal cone, normal flatness, and normal pseudoflatness Geometric equimultiplicity along a smooth subspace Zariski equimultiplicity The Hironaka-Schickhoff Theorem Geometric equimultiplicity along a general subspace Zariski equimultiplicity Normal pseudoflatness REFERENCES References - Chapter I References - Chapter II References - Appendix Chapter II References - Chapter III References - Chapter IV References - Chapter V References - Appendix Chapter V References - Chapter VI References - Chapter VII References - Chapter VIII References - Chapter IX Bibliography to the Appendix GEOMETRIC EQUIMULTIPLICITY GENERAL INDEX