Shifts in Resolutions of Multigraded Modules

Upper bounds are established on the shifts in a minimal resolution of a multigraded module. Similar bounds are given on the coefficients in the numerator of the BackelinLescot rational expression for multigraded Poincaré series. Let K be a field and S = K[x1, . . . , xn] the polynomial ring with its natural n-grading. When I is an ideal generated by monomials in the variables x1, . . . , xn, the ring R = S/I is n-graded for the induced grading. We denote by deg[y] the degree in Z of a homogeneous element y in an n-graded R-module. An n-graded finite R-module M is known to have a free resolution with ith module