BETTI NUMBERS OF MULTIGRADED MODULES

In this paper we deal with lower bounds for the Betti numbers of multigaded modules over R = k[x1,..., xd]. In general the ith Betti number of a finite length multigraded module must be at least the binomial coefficient ( d i). This is achieved only when the module in question is isomorphic to R modulo a maximal R-sequence. Otherwise the lower bound for each i is increased either by ( d-1 i) or by ( d-1 i-1). If M is an arbitrary multigraded module and s is the length of a maximal R sequence in the annihilator of M then similar inequalities hold for the Betti numbers of M with d replaced by d-s. © 1991.