Regular Sequences and Resultants Research Notes in Mathematics Volume 8 Regular Sequences and Resultants Giinter Scheja Universitdt Tubingen Tubingen, Germany Uwe Storch Ruhr- Universitdt Bochum Bochum, Germany CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business AN A K PETERS BOOK First published 2011 by A K Peters, Ltd. Published 2019 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2001 by Taylor and Francis Group, LLC CRC Press is an imprint of the Taylor <& Francis Group, an informa business ISBN-13: 978-1-56881-151-2 (hbk) This book contains information obtained from authentic and highly regarded sources. Reason- able efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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QA192 .832 2001 512.9'434-dc21 2001022483 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the Psychology Press Web site at http://www.psypress.com Preface In these notes we develop aspects of elimination by treating regular se- quences and resultants. We consider quite generally anisotropic projective spaces which gained increasing attention in the last decades. Arbitrary noetherian rings will be used as base rings, and the representation runs throughout in terms of commutative algebra, which has the positive side effect that the pattern of specialization appears fitting. The key concepts are regular sequences and complete intersections. Their use benefits by the well-developed and clear duality theory. A reading of these notes thus calls for a basic knowledge of commutative algebra and of homological methods, the features of Koszul complexes, for example. We made some effort, though, to give complete and down-to-earth proofs for those parts which depend on special constructions. More details are supplied in the following descriptions of the individual chapters. Chapter I deals with two topics which do not depend on the material of the other chapters and hence can be read independently. Sections 1 and 2 treat Kronecker's method of indeterminates. The con- cept of Kronecker extensions of a ring and its modules is introduced and developed in a general set-up. These sections also provide typical examples of the kind of methods of commutative algebra used in the notes. The second topic is the study of numerical monoids, i. e. submonoids of the additive monoid N of non-negative integers containing almost all elements of N. Such a monoid turns up in a natural way as the monoid Mon(7) = Mon(7o,..., 7n) generated by the (positive) weights 70,..., 7ra of the indeterminates TO, ..., Tn in a polynomial algebra A[T] = .A[To,..., Tn] which defines the anisotropic projective space IP7(y4.) = Proj A[T] over the commutative ring A. At first reading it suffices to look through Section 3. Chapter II contains the treatment of two of the fundamental concepts which we will use extensively in the later chapters, namely the concept of reg- ular sequences and the concepts of (relative) complete intersections and locally complete intersections. This chapter may also serve as an indepen- dent introduction into methods which have useful applications in algebraic geometry as well as in complex-analytic geometry. Basic constructions and lemmata are treated in Section 6, whereas Sec- tion 7 features specialities of the graded case. V VI Preface Crucial for what follows is the treatment of — as we call them — generic polynomials in Section 8. Generic polynomials are (not necessarily homo- V geneous) polynomials Xli/ejv UVT G Q[T] = Qpo,..., Tn], where the set N of tuples of exponents v = (i/o, • • • > vn) is a finite subset of Nn+1\{0} and where the Uv are (different) indeterminates in some polynomial ring Q over Z. The fact that a sequence F$,... ,Fr of such polynomials is regu- lar, and that the Lasker-Noether decomposition of the ideal they generate has in addition suitable properties, can be characterized by combinatorial means. This leads to the concepts of admissible arid strictly admissible se- quences of generic polynomials, which may well be of interest in other parts of algebra. As a special example we treat sequences of generic binomials. In Section 9 we prove a combinatorial characterization of algebraic in- dependence of generic Laurent polynomials in Q[T, T~1]. This will be used frequently in what follows. Chapter III presents the main case of elimination (with respect to projective spaces). In Section 10 the necessary prerequisites are briefly developed. The elimination ideal TO , part of the ideal of inertia forms T, is introduced, and the so-called main theorem of elimination is proved. In Section 11 we switch to the main case of elimination, which centres around the case of n + 1 homogeneous polynomials FQ, ..., Fn in n + 1 in- determinates TO, ..., Tn . In a first step generic sequences are treated. For many purposes it suffices to handle regular sequences of generic polynomi- als, i. e. admissible ones. In the strictly admissible case more satisfactory results are derived, including the computation of the degree of elimination. Section 12 is a substantial part of the notes. The ideal of inertia forms and the elimination ideal with respect to an arbitrary regular sequence of homogeneous polynomials FQ, ..., Fn G -A [To, • • • > Tn] are determined. The means for this are provided by an extended version of duality for graded complete intersections, which is developed and discussed in detail. Thereby the ideal of elimination is shown to be divisorial if A is an integrally closed noetherian domain, hence a principal ideal ^ 0 if A is even factorial. Chapter IV, finally, deals with the resultants. To begin with we introduce in Section 13 the resultant ideal 91 = 9l(.Fo,..., Fn) for a regular sequence FQ, ..., Fn of homogeneous polynomials of positive degrees in the polyno- mial algebra .A[To,..., Tn] over an integrally closed noetherian domain A. The ideal 9t in A is a divisorial ideal like the elimination ideal TO of the polynomials and has the same zero set as TO , but it has better functorial properties and is always principal (which is not true for TO in general). Also it suits convincingly the geometric situation: The degree of 9t defines the degree of elimination in a proper way. This is the main result of Section 14 Preface VII and is again an application of straight duality theory for regular sequences. In addition, these methods yield further characterizations of the resultant divisor. In Section 15 we use the Koszul resolution to construct a canonical 7 generator R = R(J b,..., Fn) of the resultant ideal 9l(.Fo,..., Fn), which we call the resultant. Going back to the generic situation one can define the resultant for an arbitrary sequence FQ ,..., Fn of homogeneous polynomials in -4[To,..., Tn] over any base ring A in a unique way up to sign. In the classical case with weights 70 = • • • = 7« = 1 the element R & A is the classical resultant already known in the 19th century and described, for instance, by A. Hurwitz in [19]. In the last section we present some properties of resultants which are valuable for calculations, for example the norm formula and the product formula. Supplements are added to every section. Some simply provide exercises fit- ting the context or supply further information and examples, for instance showing that inequalities are sharp or that certain hypotheses are indis- pensable. Some describe details of tools or results generally known in com- mutative algebra, but not readily available in the literature. And some add material to special points we like to emphasize but couldn't pursue in the main lines of development. Added hints to crucial arguments and other remarks, put in brackets, are sometimes elaborate, at least in those supplements serving the main text directly, where the reader should not be left alone. The comments are also detailed in supplements which follow a theme of their own and in which the reader might be interested for various reasons.