Combinatorics and commutative algebra
Author(s)
Bibliographic Information
Combinatorics and commutative algebra
（Progress in mathematics, v. 41）
Birkhäuser, c1996
2nd ed
- : us
- : pbk
Available at 97 libraries
Note
Includes bibliographical references and index
Description and Table of Contents
- Volume
-
: us ISBN 9780817638368
Description
* Stanley represents a broad perspective with respect to two significant topics from Combinatorial Commutative Algebra: 1) The theory of invariants of a torus acting linearly on a polynomial ring, and 2) The face ring of a simplicial complex * In this new edition, the author further develops some interesting properties of face rings with application to combinatorics
Table of Contents
Contents.- Preface to the Second Edition.- Preface to the First Edition.- Notation.- Background: Combinatorics.- Commutative algebra and homological algebra.- Topology.- Chapter I: Nonnegative Integral Solutions to Linear Equations: Integer stochastic matrices (magic squares).- Graded algebras and modules.- Elementary aspects of N-solutions to linear equations.- Integer stochastic matrices again.- Dimension, depth, and Cohen-Macaulay modules.- Local cohomology.- Local cohomology of the modules M phi,alpha.- Reciprocity.- Reciprocity for integer stochastic matrices.- Rational points in integer polytopes.- Free resolutions.- Duality and canonical modules.- A final look at linear equations.- Chapter II: The Face Ring of a Simplicial Complex: Elementary properties of the face ring.- f-vectors and h-vectors of complexes and multicomplexes.- Cohen-Macaulay complexes and the Upper Bound Conjecture.- Homological properties of face rings.- Gorenstein face rings.- Gorenstein Hilbert Functions.- Canonical modules of face rings.- Buchsbaum complexes.- Chapter III: Further Aspects of Face Rings: Simplicial polytopes, toric varieties, and the g-theorem.- Shellable simplicial complexes.- Matroid complexes, level complexes, and doubly Cohen-Macaulay complexes.- Balances complexes, order complexes, and flag complexes.- Splines.- Algebras with straightening law and simplical posets.- Relative simplical complexes.- Group actions.- Subcomplexes.- Subdivisions.- Problems on Simplicial Complexes and their Face Rings.- Bibliography.- Index.
- Volume
-
: pbk ISBN 9780817643690
Description
* Stanley represents a broad perspective with respect to two significant topics from Combinatorial Commutative Algebra:
1) The theory of invariants of a torus acting linearly on a polynomial ring, and
2) The face ring of a simplicial complex
* In this new edition, the author further develops some interesting properties of face rings with application to combinatorics
Table of Contents
Contents.- Preface to the Second Edition.- Preface to the First Edition.- Notation.- Background: Combinatorics.- Commutative algebra and homological algebra.- Topology.- Chapter I: Nonnegative Integral Solutions to Linear Equations: Integer stochastic matrices (magic squares).- Graded algebras and modules.- Elementary aspects of N-solutions to linear equations.- Integer stochastic matrices again.- Dimension, depth, and Cohen-Macaulay modules.- Local cohomology.- Local cohomology of the modules M phi,alpha.- Reciprocity.- Reciprocity for integer stochastic matrices.- Rational points in integer polytopes.- Free resolutions.- Duality and canonical modules.- A final look at linear equations.- Chapter II: The Face Ring of a Simplicial Complex: Elementary properties of the face ring.- f-vectors and h-vectors of complexes and multicomplexes.- Cohen-Macaulay complexes and the Upper Bound Conjecture.- Homological properties of face rings.- Gorenstein face rings.- Gorenstein Hilbert Functions.- Canonical modules of face rings.- Buchsbaum complexes.- Chapter III: Further Aspects of Face Rings: Simplicial polytopes, toric varieties, and the g-theorem.- Shellable simplicial complexes.- Matroid complexes, level complexes, and doubly Cohen-Macaulay complexes.- Balances complexes, order complexes, and flag complexes.- Splines.- Algebras with straightening law and simplical posets.- Relative simplical complexes.- Group actions.- Subcomplexes.- Subdivisions.- Problems on Simplicial Complexes and their Face Rings.- Bibliography.- Index.
- Volume
-
ISBN 9783764338367
Description
This text offers an overview of two of the main topics in the connections between commutative algebra and combinatorics. The first concerns the solutions of linear equations in non-negative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the upper bound conjecture for spheres. An introductory chapter giving background information in algebra, combinatorics and toplogy aims to broaden access to this material for non-specialists. This edition contains a chapter surveying more recent work related to face rings, focusing on applications to f-vectors. Also included is information on subcomplexes and subdivisions of simplicial complexes, and an application to spline theory.
by "Nielsen BookData"