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Graded syzygies. (English) Zbl 1213.13002
Algebra and Applications 14. London: Springer (ISBN 978-0-85729-176-9/hbk; 978-0-85729-177-6/ebook). xi, 302 p. (2011).
A syzygy of a module is a trivial relation among its generators. So it is measuring the non-freeness in terms of relations. Its iterative construction provides a chain of syzygy modules. The study of syzygies became a powerful tool at least with D. Hilbert’s work published in [Math. Ann. XXXVI. 473–534 (1890; JFM 22.0133.01), and Math. Ann. XLII. 313–373 (1893; JFM 25.0173.01)]. He proved that the chain of syzygies of a finitely generated graded \(S\)-module (\(S = k[x_1,\dots,x_n]\)) becomes trivial after at most \(n\) steps.
This was extended to regular local rings by Serre resp. Auslander and Buchsbaum (see [J.-P. Serre, Algèbre locale. Multiplicités. Cours au Collège de France, 1957–1958, Lecture Notes in Mathematics. 11. Berlin-Heidelberg-New York: Springer-Verlag (1965; Zbl 0142.28603)] and the references there) with the concept of projective dimension and the characterization of regular local rings by the finiteness of the projective dimension of the residue field. More recently the applications and the study of syzygies plays an essential rôle in algebraic geometry e.g. with the notion of Castelnuovo-Mumford regularity and related problems. An extensive study of these new phenomeneons is presented in David Eisenbud’s book, see [D. Eisenbud, The geometry of syzygies. A second course in commutative algebra and algebraic geometry. Graduate Texts in Mathematics 229. New York, NY: Springer (2005; Zbl 1066.14001)].
While Hilbert himself used the canonical grading of \(S\) this is the most important feature for applications in algebraic geometry. So the higher syzygies are finitely generated graded \(S\)-modules for the defining ideal of a projective variety. That is, the fine structure of an algebraic variety might be determined by the graded structure of the higher syzygy modules.
This is the starting point for the paper under review, published as Volume 14 in the Springer Series Algebra and Applications, a series that aims to publish up-to-date information about progress in all fields of algebra. The intention of the present book is – as the title announces – the study of graded syzygies not only for the use in algebraic geometry but also from the point of view of commutative algebra starting from the beginning of the theory.
The book is divided into four Chapters with 74 subsections. The first chapter “Graded free resolutions” contains the basic material on graded free resolutions. It covers half of the content of the book. The author presents in an elegant and condensed way the basics including tools from homological algebra, Gröbner bases, Koszul rings etc. All that is illustrated by examples and problems. The second chapter “Hilbert functions” provides the study of Hilbert functions via Macaulay’s idea of lex ideals. That is, for every graded ideal there exists a lex ideal with the same Hilbert function. Moreover, the chapter covers Green’s Theorem, Gotzmann’s Persistence and Gotzmann’s Regularity Theorem as well as the Eisenbud-Green-Harris conjecture.
Any graded ideal \(I \subset S\) has the Hilbert function of a monomial ideal. The Hilbert function is determined by the minimal free resolution. Therefore it is of some interest to investigate minimal free resolutions of monomial ideals. This is the subject of Chapter 3 “Monomial Resolutions”. In general there is no guide to understand the minimal free resolutions of monomial ideals. The Chapter covers – among others – results about the Scarf complex, Lyubeznik’s resolution and in particular the description of resolutions via the Stanlay-Reisner correspondance by topological methods. In the final Chapter “Syzygies of Toric Ideals” there is the application of some of the previous ideas to the study of the free resolution of toric ideals by using multi-gradings.
A valuable feature of this monograph is the inclusion of open problems and conjectures. They guide an interested reader to certain research directions in the field. With its clear and self-contained exposition the book is intended for graduate students as an introduction to the field with perspectives for further research. It can also be used to senior mathematicians for an information about the subject. The reviewer welcomes the book as a perfect start for understanding the problems around syzygies. It shows that syzygies play a central rôle in present day research in commutative algebra.

13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13D02 Syzygies, resolutions, complexes and commutative rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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