The structure of Gorenstein-linear resolutions of Artinian algebras

Abstract

Let k be a field, A a standard-graded Artinian Gorenstein k-algebra, S the standard-graded polynomial ring Sym•kA1, I the kernel of the natural map ▪, d the vector space dimension dimk⁡A1, and n the least index with In≠0. Assume that 3≤d and 2≤n. In this paper, we give the structure of the minimal homogeneous resolution B of A by free S-modules, provided B is Gorenstein-linear. (Keep in mind that if A has even socle degree and is generic, then A has a Gorenstein-linear minimal resolution.)Our description of B depends on a fixed, but arbitrary, decomposition of A1 of the form kx1⊕V0, for some non-zero element x1 and some (d−1) dimensional subspace V0 of A1. Much information about B is already contained in the complex B‾=B/x1B, which we call the skeleton of B. One striking feature of B is the fact that the skeleton of B is completely determined by the data (d,n); no other information about A is used in the construction of B‾.The skeleton B‾ is the mapping cone of zero:K→L, where L is a well known resolution of Buchsbaum and Eisenbud; K is the dual of L; and L and K are comprised of Schur and Weyl modules associated to hooks, respectively. The decomposition of B‾ into Schur and Weyl modules lifts to a decomposition of B; furthermore, B inherits the natural self-duality of B‾.The differentials of B are explicitly given, in a polynomial manner, in terms of the coefficients of a Macaulay inverse system for A. In light of the properties of B‾, the description of the differentials of B amounts to giving a minimal generating set of I, and, for the interior differentials, giving the coefficients of x1. As an application we observe that every non-zero element of A1 is a weak Lefschetz element for A.