Table Of ContentM. 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
Manfred Herrmann
Ulrich Orbanz
Mathematisches Institut der UniversitiH zu K61n
Weyertal 86-90,0-5000 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: 978-3-642-64803-8 e-IS8N-13: 978-3-642-61349-4
001: 10.1007/978-3-642-61349-4
Libary of Congress Cataloging-in-Publication Data. Herrmann, Manfred, 1932-.
Equimultiplicity and blowing up. Bibliography: p. Includes index. 1. Multiplicity
(Mathematics) 2. Blowing up (Algebraic geometry) 3. Local rings. I. Ikeda, S. (Shin),
1948-.11. Orbanz, Ulrich, 1945-.111. Title. QA251.38.H471988 512 88-4660
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© Springer-Verlag Berlin Heidelberg 1988
Softcover reprint of the hardcover 1s t edition 1988
2141/3140-543210
To GeJr..cU
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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 con-
ditions 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 pro
cedure 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.
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 normaliza
tions.
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 genera
lized 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 geome
try is that they furnish some way of measuring and comparing singula
rities. 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 multi
plicity. 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 multipli
cities.
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 ge
nerated 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
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 quasi
unmixed local rings are given. A very useful tool for these characteri
zations 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 generaliza
tion: 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 equi
multiplicity together with a numerical description of each condition.
Then we describe the hierarchy among these conditions. Finally we in
vestigate these conditions concerning openess and transitivity proper
ties.
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 semiconti
nuity one has to desingularize curves by blowing up points, and conver
sely for inequalities of Hilbert functions under blowing up other
centers one has to use semicontinuity.
Chapter VII, presupposing the following Chapters VIII and IX, dis
cusses local cohomology and duality over graded rings. For local rings,
the theory of local duality can be found in textbooks. For the corres
ponding results over graded rings we give detailed proofs because they
x
are not all out available in the literature.
Chapter VIII studies local rings (A,m) with finite local cohomo
logy 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 multi
plicities. 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 Divi
sion 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 deve
lops the theory of compact Stein neighbourhoods further and thus de
duces 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 re
sults 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 ex
press our gratitude in particular to J. Giraud, J. Lipman, R. Sharp
and J.L. Vicente for suggestions and encouragements during the pre
paration 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
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 Mathe
matics 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.
Table of contents
Chapter I - REVIEW OF MULTIPLICITY THEORY ..................... .
§ The multiplicity symbol .............................. 1
§ 2 Hilbert functions ................•................... 6
§ 3 Generalized reultiplicities and Hilbert functions ..... 10
§ 4 Reductions and integral closure of ideals ............ 16
§ 5 Faithfully flat extensions ........•.•................ 25
§ 6 Projection formula and criterion for
multiplicity one ..................................... 27
§ 7 Examples ........................•..................•. 34
Chapter II - Z-GRADED RINGS AND MODULES ........................ 44
§ 8 Associated graded rings and Rees algebras ....•.....•. 44
§ 9 Dimension............................................ 49
§10 Homogeneous parameters ............................... 55
§11 Regular sequences on graded modules •................. 68
§ 12 Review on blowing up ............................•.... 77
§ 13 Standard bases .....................•................. 88
§14 Examples 100
Appendix - Homogeneous subrings of a homogeneous ring .......... 112
Chapter III - ASYMPTOTIC SEQUENCES AND QUASI-UNMIXED RINGS ..... 117
§15 Auxiliary results on integral dependence of ideals .•. 117
§16 Associated primes of the integral closure of
powers of an ideal .................. ". ................ 122
§ 17 Asymptotic sequences ................................. 133
§18 Quasi-unmixed rings .................................. 137
§19 The theorem of Rees-Boger ............................ 146
Chapter IV - VARIOUS NOTIONS OF EQUIMULTIPLE AND
PERMISSIBLE IDEALS ................................ 152
§20 Reinterpretation of the theorem of Rees-Boger ........ 152
§21 Hironaka-Grothendieck homomorphism ................... 159
§22 Projective normal flatness and numerical
characterization of permissibility ..............•..•. 166
§23 Hierarchy of equimultiplicity and permissibility ..... 182
§24 Open conditions and transitivity properties .......... 194
