Castelnuovo-Mumford regularity and projective dimension of a squarefree monomial ideal

Abstract

Let S = K[x₁, x₂, . . . , x_n] be the polynomial ring in n variables over a field K, and let I be a squarefree monomial ideal minimally generated by the monomials u₁, u₂, . . . , u_m. Let w be the smallest number t with the property that for all integers such that lcm (u_{i_1},u_{i_2},...,u_{i_t})=lcm(u₁, u₂, . . . , u_m). We give an upper bound for Castelnuovo-Mumford regularity of I by the bigsize of I. As a corollary, the projective dimension of I is bounded by the number w.

Key words： Castelnuovo-Mumford regularity; projective dimension; squarefree monomial ideals

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