Graduate Texts in Mathematics 229. New York, NY: Springer (ISBN 0-387-22232-4/pbk; 0-387-22215-4/hbk). xvi, 243 p. $ 29.95, EUR 29.95/net, £ 23.00, sFr 54.40/pbk; $ 64.15/hbk (2005).
In his introduction to the French edition of F. Klein’s Erlanger Programm [Le programm d’Erlangen (1974; Zbl 0282.50012)], J. Dieudonne wrote:
“The developments which most will have influence on Klein come from years 1850–1860...we have the construction, due to Cayley and Sylvester, of the general Theory of Invariants, which will offer soon a procedure [the symbolic method ] to determine the algebraic invariants of a system of geometric object and all the algebraic relations [or syzygies] among them".
So when the new way to look at geometry, or better at geometries, enters the game at the end of XIX century (a geometry is the study of certain objects and a group of transformations among them, as Klein defines it in his program), the study of syzygies is already part of this new approach, also if it will be with Hilbert that its role will become clearer and the approach to syzygy modules more structured.
As the author points out at the beginning of the book, when you study geometry using the tools given by gbstract algebra one of the main problems is how to relate equations and geometric objects, i.e. how to extract the geometric information from the equations defining such objects. The theory of syzygies is one powerful tool to do this (a microscope, Eisenbud says).
This book is devoted to offer, as the title says already, an approach to the study of this algebraic subject (syzygy = relation among generators of a module) with a very geometry-oriented point of view which is actually consubstantial with the way the theory has been born. The exposition is at a graduate level, and a student would learn a lot of (classical and less classical) algebraic geometry from it.
The double bet of the book is to be able to be a complete textbook for a (second level) graduate course in algebraic geometry or commutative algebra and at the same time to become a useful reference text for research work on the subject. I would say that both aspects of the bet have been gained, since the book manages to be introductory but also gives a fairly complete outlook on the most recent result about syzygies and their applications to algebraic geometry. It also gives a good idea of how the theory has been developed and what are the questions that have pushed towards the main results.
Every chapter begins with an informal sketch of what will follow, giving motivations for the aspects of the theory to be developed, and geometric examples and application will spread light on the algebraic construction.
After a nice preface that gives an idea of the subject, the reader is introduced to Hilbert functions and free resolutions (chapter one) immediately followed in chapters 2 and 3 by examples (monomial ideals, ideals of sets of points in and ) which shows how syzygies of ideals can carry information about geometric configurations.
Castelnuovo-Mumford regularity is introduced in chapter 4, and the problem of interpolation along with the cohomological point of view are again intertwined along the chapter so to give motivations and examples along with the development of the theory, which is applied in Chapter 5 to study regularity of projective curves, giving the main results (as the Green-Lazarsfeld-Peskine theorem) and conjectures on the subject at the present state of the art.
In chapter 6 embeddings via linear systems are studied and expecially matrices of linear forms (and determinantal ideals) coming from them. Here rational and elliptic normal curves are treated in detail.
A new tool is introduced in chapter 7: Bernard-Gelfand-Gelfand correspondence, and exterior algebra is here studied to give Green’s linear syzygy theorem. All these and new tools are used in chapter 8, where high degree embeddings of curves are studied, where the classical results on the subject and the Green and Green-Lazarsfeld conjecture are given. Recent results on these conjectures are described in chapter 9.
Eventually, two appendices give an useful “reminder" of local cohomology and of commutative algebra (the author prides himself that this is the shortest compendium of commutative algebra, after he wrote the longest one [see “Commutative algebra. With a view toward algebraic geometry”, Graduate Texts in Mathematics 150 (1995; Zbl 0819.13001)]).
As a final note, I would like to add that the author manages in this book to give an example of how to overcome what Simone Weil (the great sister of André) had said pointing out algebra as one of the “bad gift" of contemporary age, since algebra’s procedures can be so abstract to hide to people working with them what they are really doing. Sometimes going through an algebra text (specially the very Bourbaki-oriented ones) one could agree with her, but geometry can furnish a way out (not the only one) of this by giving “life" to algebra structures that could otherwise appear as diamonds: perfect and sterile.