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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.
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