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2013 | Book

A Survey on Stanley Depth Read chapter Read first chapter

Monomial Ideals, Computations and Applications

Editors: Anna M. Bigatti, Philippe Gimenez, Eduardo Sáenz-de-Cabezón

Publisher: Springer Berlin Heidelberg

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This work covers three important aspects of monomials ideals in the three chapters "Stanley decompositions" by Jürgen Herzog, "Edge ideals" by Adam Van Tuyl and "Local cohomology" by Josep Álvarez Montaner. The chapters, written by top experts, include computer tutorials that emphasize the computational aspects of the respective areas. Monomial ideals and algebras are, in a sense, among the simplest structures in commutative algebra and the main objects of combinatorial commutative algebra. Also, they are of major importance for at least three reasons. Firstly, Gröbner basis theory allows us to treat certain problems on general polynomial ideals by means of monomial ideals. Secondly, the combinatorial structure of monomial ideals connects them to other combinatorial structures and allows us to solve problems on both sides of this correspondence using the techniques of each of the respective areas. And thirdly, the combinatorial nature of monomial ideals also makes them particularly well suited to the development of algorithms to work with them and then generate algorithms for more general structures.

Table of Contents

Frontmatter

Stanley Decompositions

Frontmatter
A Survey on Stanley Depth
Abstract
At the MONICA conference “MONomial Ideals, Computations and Applications” at the CIEM, Castro Urdiales (Cantabria, Spain) in July 2011, I gave three lectures covering different topics of Combinatorial Commutative Algebra: (1) A survey on Stanley decompositions. (2) Generalized Hibi rings and Hibi ideals. (3) Ideals generated by two-minors with applications to Algebraic Statistics. In this article I will restrict myself to give an extended presentation of the first lecture. The CoCoA tutorials following this survey will deal also with topics related to the other two lectures. Complementing the tutorials, the reader finds in [165] a CoCoA routine to compute the Stanley depth for modules of the form I ∕ J, where J ⊂ I are monomial ideals.
Jürgen Herzog
Stanley Decompositions Using CoCoA
Abstract
First released in 1988, CoCoA is a special-purpose Computer Algebra System for doing Computations in Commutative Algebra. It is freely available and offers a textual interface, an Emacs mode, and a graphical user interface common to most platforms [39].
Anna Maria Bigatti, Emanuela De Negri

Edge Ideals

Frontmatter
A Beginner’s Guide to Edge and Cover Ideals
Abstract
Monomial ideals, although intrinsically interesting, play an important role in studying the connections between commutative algebra and combinatorics. Broadly speaking, problems in combinatorics are encoded into monomial ideals, which then allow us to use techniques and methods in commutative algebra to solve the original question. Stanley’s proof of the Upper Bound Conjecture [180] for simplicial spheres is seen as one of the early highlights of exploiting this connection between two fields. To bridge these two areas of mathematics, Stanley used square-free monomial ideals.
Adam Van Tuyl
Edge Ideals Using Macaulay2
Abstract
Computer algebra systems, like Macaulay 2 [80], Singular [47], and CoCoA [39], have become essential tools for many mathematicians in commutative algebra and algebraic geometry. These systems provide a “laboratory” in which we can experiment and play with new ideas. From these experiments, a researcher can formulate new conjectures, and hopefully, new theorems. Computer algebra systems are especially good at dealing with monomial ideals. As a consequence, the study of edge and cover ideals is well suited to experiments using computer algebra systems.
Adam Van Tuyl

Local Cohomology

Frontmatter
Local Cohomology Modules Supported on Monomial Ideals
Abstract
Local cohomology was introduced by A. Grothendieck in the early 1960s and quickly became an indispensable tool in Commutative Algebra. Despite the effort of many authors in the study of these modules, their structure is still quite unknown. C
Josep Àlvarez Montaner
Local Cohomology Using Macaulay2
Abstract
Over the last 20 years there were many advances made in the computational theory of D-modules. Nowadays, the most common computer algebra systems such as Macaulay2 or Singular have important available packages for working with D-modules. In particular, the package D-modules [127] for Macaulay 2 [80] developed by A. Leykin and H. Tsai contains an implementation of the algorithms given by U. Walther [194] and T. Oaku and N.
Josep Àlvarez Montaner, Oscar Fernández-Ramos
Backmatter
Metadata
Title
Monomial Ideals, Computations and Applications
Editors
Anna M. Bigatti
Philippe Gimenez
Eduardo Sáenz-de-Cabezón
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-38742-5
Print ISBN
978-3-642-38741-8
DOI
https://doi.org/10.1007/978-3-642-38742-5

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