Combinatorics and Commutative Algebra

Volume 41

Richard P. Stanley - Collection Progress in Mathematics

175 pages, parution le 08/07/2005

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Résumé

Some remarkable connections between commutative algebra and combinatorics have been discovered in recent years. This book provides an overview of two of the main topics in this area. The first concerns the solutions of linear equations in nonnegative 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 topology broadens access to this material for non-specialists.

New to this edition is a chapter surveying more recent work related to face rings, focusing on applications to f-vectors.

Sommaire

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

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	PAPIER
Éditeur(s)	Birkhäuser
Auteur(s)	Richard P. Stanley
Collection	Progress in Mathematics
Parution	08/07/2005
Nb. de pages	175
Format	15,5 x 23
Couverture	Broché
Poids	270g
Intérieur	Noir et Blanc
EAN13	9780817643690
ISBN13	978-0-81764-369-0