Provides an overview of connections between commutative algebra and combinatorics. Topics include linear equations in nonnegative integers; integer stochastic matrices; the volume of polytopes; combinatorial reciprocity theorems; and the face ring of a simplicial complex. Features problems and an introductory chapter on algebra, combinatorics, and topology. This second edition contains a chapter surveying recent work in face rings and simplicial complexes, and an application to spline theory. Annotation c. by Book News, Inc., Portland, Or.
* 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
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.
