For every mathematician, ring theory and K-theory are intimately connected: al- braic K-theory is largely the K-theory of rings. At ?rst sight, polytopes, by their very nature, must appear alien to surveyors of this heartland of algebra. But in the presence of a discrete structure, polytopes de?ne a?ne monoids, and, in their turn, a?ne monoids give rise to monoid algebras. Teir spectra are the building blocks of toric varieties, an area that has developed rapidly in the last four decades. From a purely systematic viewpoint, "monoids" should therefore replace "po- topes" in the title of the book. However, such a change would conceal the geometric ?avor that we have tried to preserve through all chapters. Before delving into a description of the contents we would like to mention three general features of the book: (?) the exhibiting of interactions of convex geometry, ring theory, and K-theory is not the only goal; we present some of the central results in each of these ?elds; (?) the exposition is of constructive (i. e., algorithmic) nature at many places throughout the text-there is no doubt that one of the driving forces behind the current popularity of combinatorial geometry is the quest for visualization and computation; (? ) despite the large amount of information from various ?elds, we have strived to keep the polytopal perspective as the major organizational principle.

The central theme of this volume is commutative algebra, with emphasis on special graded algebras, which are increasingly of interest in problems of algebraic geometry, combinatorics and computer algebra. Most of the papers have partly survey character, but are research-oriented, aiming at classification and structural results.

Determinantal rings and varieties have been a central topic of commutative algebra and algebraic geometry. Their study has attracted many prominent researchers and has motivated the creation of theories which may now be considered part of general commutative ring theory. The book gives a first coherent treatment of the structure of determinantal rings. The main approach is via the theory of algebras with straightening law. This approach suggest (and is simplified by) the simultaneous treatment of the Schubert subvarieties of Grassmannian. Other methods have not been neglected, however. Principal radical systems are discussed in detail, and one section is devoted to each of invariant and representation theory. While the book is primarily a research monograph, it serves also as a reference source and the reader requires only the basics of commutative algebra together with some supplementary material found in the appendix. The text may be useful for seminars following a course in commutative ring theory since a vast number of notions, results, and techniques can be illustrated significantly by applying them to determinantal rings.

This book offers an up-to-date, comprehensive account of determinantal rings and varieties, presenting a multitude of methods used in their study, with tools from combinatorics, algebra, representation theory and geometry. After a concise introduction to Groebner and Sagbi bases, determinantal ideals are studied via the standard monomial theory and the straightening law. This opens the door for representation theoretic methods, such as the Robinson-Schensted-Knuth correspondence, which provide a description of the Groebner bases of determinantal ideals, yielding homological and enumerative theorems on determinantal rings. Sagbi bases then lead to the introduction of toric methods. In positive characteristic, the Frobenius functor is used to study properties of singularities, such as F-regularity and F-rationality. Castelnuovo-Mumford regularity, an important complexity measure in commutative algebra and algebraic geometry, is introduced in the general setting of a Noetherian base ring and then applied to powers and products of ideals. The remainder of the book focuses on algebraic geometry, where general vanishing results for the cohomology of line bundles on flag varieties are presented and used to obtain asymptotic values of the regularity of symbolic powers of determinantal ideals. In characteristic zero, the Borel-Weil-Bott theorem provides sharper results for GL-invariant ideals. The book concludes with a computation of cohomology with support in determinantal ideals and a survey of their free resolutions. Determinants, Groebner Bases and Cohomology provides a unique reference for the theory of determinantal ideals and varieties, as well as an introduction to the beautiful mathematics developed in their study. Accessible to graduate students with basic grounding in commutative algebra and algebraic geometry, it can be used alongside general texts to illustrate the theory with a particularly interesting and important class of varieties.

For every mathematician, ring theory and K-theory are intimately connected: al- braic K-theory is largely the K-theory of rings. At ?rst sight, polytopes, by their very nature, must appear alien to surveyors of this heartland of algebra. But in the presence of a discrete structure, polytopes de?ne a?ne monoids, and, in their turn, a?ne monoids give rise to monoid algebras. Teir spectra are the building blocks of toric varieties, an area that has developed rapidly in the last four decades. From a purely systematic viewpoint, "monoids" should therefore replace "po- topes" in the title of the book. However, such a change would conceal the geometric ?avor that we have tried to preserve through all chapters. Before delving into a description of the contents we would like to mention three general features of the book: (?) the exhibiting of interactions of convex geometry, ring theory, and K-theory is not the only goal; we present some of the central results in each of these ?elds; (?) the exposition is of constructive (i. e., algorithmic) nature at many places throughout the text-there is no doubt that one of the driving forces behind the current popularity of combinatorial geometry is the quest for visualization and computation; (? ) despite the large amount of information from various ?elds, we have strived to keep the polytopal perspective as the major organizational principle.

In the past two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the subject. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. The general theory is applied to a number of examples and the connections with combinatorics are highlighted. Throughout each chapter, the authors have supplied many examples and exercises.