Syzygies and Hilbert Functions

Irena Peeva

March 20, 2007 by Chapman and Hall/CRC
Reference - 304 Pages - 6 B/W Illustrations
ISBN 9781584888604 - CAT# C8601
Series: Lecture Notes in Pure and Applied Mathematics

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  • Presents highlights, conjectures, unsolved problems, and examples of Hilbert functions and resolutions
  • Covers topics at the interface of commutative algebra, algebraic geometry, and combinatorics
  • Discusses the important invariant of Castelnuovo-Mumford regularity
  • Surveys two challenging conjectures: the LPP conjecture and the multiplicity conjecture
  • Describes bigraded rings, multigraded rings, and toric rings
  • Summary

    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.