Note on bounds for multiplicities

Abstract

Let S = K[x(1),...,x(n)] be a polynomial ring and R = S/I be a graded K-algebra where I subset of S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R) = 2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals. (C) 2004 Elsevier B.V. All rights reserved.