Preface
The last decade has seen a number of exciting developments at the intersection of commutative algebra with combinatorics. New methods have evolved out of an inux of ideas from such diverse areas as polyhedral geometry, theoretical physics, representation theory, homological algebra, symplectic geometry, graph theory, integer programming, symbolic computation, and statistics. The purpose of this volume is to provide a selfcontained introduction to some of the resulting combinatorial techniques for dealing with polynomial rings, semigroup rings, and determinantal rings.
Our exposition mainly concerns combinatorially de_ned ideals and their quotients, with a focus on numerical invariants and resolutions, especially under gradings more re_ned than the standard integer grading.
This project started at the COCOA summer school in Torino, Italy, in
June 1999. The eight lectures on monomial ideals given there by Bernd
Sturmfels were later written up by Ezra Miller and David Perkinson and published in [MP01]. We felt it would be nice to add more material and turn the COCOA notes into a real book. What you hold in your hand is the result, with Part I being a direct outgrowth of the COCOA notes.
Combinatorial commutative algebra is a broad area of mathematics, and one can cover but a small selection of the possible topics in a single book.
Our choices were motivated by our research interests and by our desire to reach a wide audience of students and researchers in neighboring _elds.
Numerous references, mostly con_ned to the Notes ending each chapter, point the reader to closely related topics that we were unable to cover.
A milestone in the development of combinatorial commutative algebra was the 1983 book by Richard Stanley [Sta96]. That book, now in its second edition, is still an excellent source. We have made an attempt to complement and build on the material covered by Stanley. Another boon to the subject came with the arrival in 1995 of the book by Bruns and Herzog
[BH98], also now in its second edition. The middle part of that book, on
\Classes of Cohen{Macaulay rings", follows a progression of three chapters on combinatorially de_ned algebras, from Stanley{Reisner rings through semigroup rings to determinantal rings. Our treatment elaborates on these three themes. The inuence of [BH98] can seen in the subdivision of our book into three parts, following the same organizational principle. vii viii PREFACE
We frequently refer to two other textbooks in the same Springer series as ours, namely Eisenbud's book on commutative algebra [Eis95] and Ziegler's book on convex polytopes [Zie95]. Students will _nd it useful to place these two books next to ours on their shelves. Other books in the GTM series that contain useful material related to combinatorial commutative algebra are
[BB04], [Eis04], [EH00], [Ewa96], [Gru03], [Har77], [MacL98], and [Rot88].
There are two other _ne books that o_er an introduction to combinatorial commutative algebra from a perspective di_erent than ours, namely the ones by Hibi [Hib92] and Villarreal [Vil01]. Many readers of our book will enjoy learning more about computational commutative algebra as they go along; for this we recommend the books by Cox, Little, and
O'Shea [CLO98], Greuel and P_ster [GP02], Kreuzer and Robbiano [KR00],
Schenck [Sch03], Sturmfels [Stu96], and Vasconcelos [Vas98]. Additional material can be found in the proceedings volumes [EGM98] and [AGHSS04].
Drafts of this book have been used for graduate courses taught by Victor
Reiner at the University of Minnesota and by the authors at UC Berkeley.
In our experience, covering all 18 chapters would require a full-year course, either two semesters or three quarters (one for each of Part I, Part II, and
Part III). For a _rst introduction, we view Chapter 1 and Chapters 3{8 as being essential. However, we recommend that this material be supplemented with a choice of one or two of the remaining chapters, to get a feel for a speci_c application of the theory presented in Chapters 7 and 8. Topics that stand alone well for this purpose are Chapter 2 (which could, of course, be presented earlier), Chapter 9, Chapter 10, Chapter 11, Chapter 14, and
Chapter 18. We have also observed success in covering Chapter 12 with only the barest introduction to injective modules from Chapter 11, although
Chapters 11 and 12 work even more coherently as a pair. Other two-chapter sequences include Chapters 11 and 13 or Chapters 15 and 16. Although the latter pair forms a satisfying end, it becomes even more so as a triplet with
Chapter 17. Advanced courses could begin with Chapters 7 and 8 and continue with the rest of Part II, or instead continue with Part III.
In general, we assume knowledge of commutative algebra (graded rings, free resolutions, Grobner bases, and so on) at a level on par with the undergraduate textbook of Cox, Little, and O'Shea [CLO97], supplemented with a little bit of simplicial topology and polyhedral geometry. Although these prerequisites are fairly modest, the mix of topics calls for considerable mathematical maturity. Also, more will be gained from some of the later chapters with additional background in homological algebra or algebraic geometry. For the former, this is particularly true of Chapters 11 and 13, whereas for the latter, we are referring to Chapter 10 and Chapters 15{18.
Often we work with algebraic groups, which we describe explicitly by saying what form the matrices have (such as \block lower-triangular"). All of our arguments that use algebraic groups are grounded _rmly in the transparent linear algebra that they represent. Typical conclusions reached using algebraic geometry are the smoothness and irreducibility of orbits. Typical
PREFACE ix uses of homological algebra include statements that certain operations (on resolutions, for example) are well-de_ned independent of the choices made.
Each chapter begins with an overview and ends with Notes on references and pointers to the literature. Theorems are, for the most part, attributed only in the Notes. When an exercise is based on a speci_c source, that source is credited in the Notes. For the few exercises used in the proofs of theorems in the main body of the text, solutions to the nonroutine ones are referenced in the Notes. The References list the pages on which each source is cited. The mathematical notation throughout the book is kept as consistent as possible, making the glossary of notation particularly handy, although some of our standard symbols occasionally moonlight for brief periods in nonstandard ways, when we run out of letters. Cross-references have the form \Item aa.bb" if the item is number bb in Chapter aa. Finally, despite our best e_orts, errors are sure to have kept themselves safely hidden from our view. Please do let us know about all the bugs you may discover.
In August 2003, a group of students and postdocs ran a seminar at
Berkeley covering topics from all 18 chapters. They read the manuscript carefully and provided numerous comments and improvements. We wish to express our sincere gratitude to the following participants for their help:
Matthias Beck, Carlos D'Andrea, Mike Develin, Nicholas Eriksson, Daniel
Giaimo, Martin Guest, Christopher Hillar, Serkan Ho_sten, Lionel Levine,
Edwin O'Shea, Julian Pfeie, Bobby Poon, Nicholas Proudfoot, Brian
Rothbach, Nirit Sandman, David Speyer, Seth Sullivant, Lauren Williams,
Alexander Woo, and Alexander Yong. Additional comments and help were provided by David Cox, Alicia Dickenstein, Jesus De Loera, Joseph Gubeladze,
Mikhail Kapranov, Diane Maclagan, Raymond Hemmecke, Bjarke
Roune, Olivier Ruatta, and Gunter Ziegler. Special thanks are due to Victor
Reiner, for the many improvements he contributed, including a number of exercises and corrections of proofs. We also thank our coauthors Dave
Bayer, Mark Haiman, David Helm, Allen Knutson, Misha Kogan, Laura
Matusevich, Isabella Novik, Irena Peeva, David Perkinson, Sorin Popescu,
Alexander Postnikov, Mark Shimozono, Uli Walther, and Kohji Yanagawa, from whom we have learned so much about combinatorial commutative algebra, and whose contributions form substantial parts of this book.
A number of organizations and nonmathematicians have made this book possible. Both authors had partial support from the National Science Foundation.
Ezra Miller was a postdoctoral fellow at MSRI Berkeley in 2003.
Bernd Sturmfels was supported by the Miller Institute at UC Berkeley in
2000{2001, and as a Hewlett{Packard Research Professor at MSRI Berkeley in 2003{2004. Our editor, Ina Lindemann, kept us on track and helped us to _nish at the right moment. Most of all, we thank our respective partners,
Elen and Hyungsook, for their boundless encouragement and support.
Ezra Miller, Minneapolis, MN ezra@math.umn.edu
Bernd Sturmfels, Berkeley, CA bernd@math.berkeley.edu
12 May 2004