دانلود کتاب Combinatorics of Determinantal Ideals

The study of determinantal ideals and of classical determinantal rings is an old topic of commutative algebra. As in most of the cases, the theory evolved from algebraic geometry, and soon became an important topic in commutative algebra. Looking back, one can say that it is the merit of Eagon and Northcott to be the first who brought to the attention of algebraists the determinantal ideals and investigated them by the methods of commutative and homological algebra. Later on, Buchsbaum and Eisenbud, in a series of articles, went further along the way of homological investigation of determinantal ideals, while Eagon and Hochster studied them using methods of commutative algebra in order to prove that the classical determinantal rings are normal Cohen-Macaulay domains. Subsequently, L. Avramov, T. Jozefiak, H. Kleppe, R. E. Kutz, D. Laksov, V. Marinov, P. Pragacz et al. have turned the attention to the study of determinantal (pfaffian) ideals, respectively of classical determinantal (pfaffian) rings associated to symmetric (alternating) matrices. As shown later by C. De Concini, D. Eisenbud, and C. Procesi, the appropriate framework including all three types of rings is that of algebras with straightening law, and the standard monomial theory on which these algebras are based yields computationally effective results. A coherent treatment of determinantal ide- als from this point of view was given by Bruns and Vetter in their seminal book.

A new perspective to the study of the determinantal ideals was brought by Sturmfels’ paper, in which he applied the Knuth-Robinson-Schensted correspondence in order to determine Gröbner bases for determinantal ideals. Later on, this technique was extended by Herzog an Trung to the study of 1-cogenerated ideals, ladder determinantal ideals and pfaffian ideals.

Our book aims to a thorough treatment of all three types of determinantal rings in the light of the achievements of the last fifteen years since the publication of Bruns and Vetter’s book. We implicitly assume that the reader is familiar with the basics of commutative algebra. However, we include some of the main notions and results from Bruns and Vetter, for the sake of completeness, but without proofs. We recommend the reader to first look at the book of Bruns and Vetter in order to get a feel for the flavor of this field. The first two chapters of our book follow the line of investigation from Bruns and Vetter, while the rest of the book relies on the combinatorics of Gröbner bases, a method initiated by Sturmfels. As often as possible, we discuss the case of 1-cogenerated ideals and their corresponding determinantal rings, though sometimes we restrict ourselves to the classical case of determinantal ideals generated by minors (pfaffians) of fixed size, either due to lack of results or difficulties of exposition.

An important and useful feature of the book is that every chapter contains a “Notes” section, where we present historical remarks, more bibliographical sources, open questions and further directions for research.
Our book is meant to be a reference text for the current state of research in the theory of determinantal rings. It was structured in such a way that it can be used as textbook for a one semester graduate course in advanced topics in Commutative Algebra, at PhD level.