El. knyga: Syzygies and Hilbert Functions

Primarily drawn from an October 2005 conference held at Cornell, ten papers present recent results on Hilbert functions and free resolutions. The longest papers explore questions about Castelnuovo-Mumford regularity and illustrate how bigraded commutative algebra differs from the classical graded case. Other topics include Hilbert coefficients of ideals, Lex-plus-powers ideals, multiplicity conjectures, the geometry of Hilbert functions, infinite free resolutions over toric rings, and subspace arrangements. Annotation ©2007 Book News, Inc., Portland, OR (booknews.com)

Hilbert functions and resolutions are both central objects in commutative algebra and fruitful tools in the fields of algebraic geometry, combinatorics, commutative algebra, and computational algebra. Spurred by recent research in this area, Syzygies and Hilbert Functions explores fresh developments in the field as well as fundamental concepts.

Written by international mathematics authorities, the book first examines the invariant of Castelnuovo-Mumford regularity, blowup algebras, and bigraded rings. It then outlines the current status of two challenging conjectures: the lex-plus-power (LPP) conjecture and the multiplicity conjecture. After reviewing results of the geometry of Hilbert functions, the book considers minimal free resolutions of integral subschemes and of equidimensional Cohen-Macaulay subschemes of small degree. It also discusses relations to subspace arrangements and the properties of the infinite graded minimal free resolution of the ground field over a projective toric ring. The volume closes with an introduction to multigraded Hilbert functions, mixed multiplicities, and joint reductions.

By surveying exciting topics of vibrant current research, Syzygies and Hilbert Functions stimulates further study in this hot area of mathematical activity.