Free resolutions for multigraded modules: a generalization of Taylor’s construction

Abstract

Let Q = k(x1,... ,xn) be a polynomial ring over a field k with the standard Nn-grading. Letbe a morphism of finite free Nn-graded Q- modules. We translate to this setting several notions and constructions that appear originally in the context of monomial ideals. First, using a modification of the Buchsbaum-Rim complex, we construct a canonical complex T•(�) of finite free N n -graded Q-modules that generalizes Taylor's resolution. This complex provides a free resolution for the cokernel M ofwhensatisfies certain rank criteria. We also introduce the Scarf complex of �, and a notion of generic morphism. Our main result is that the Scarf complex ofis a minimal free resolution of M whenis minimal and generic. Finally, we introduce the LCM-lattice forand establish its significance in determining the minimal resolution of M.