The structure of Gorenstein-linear resolutions of Artinian algebras

Abstract

This is the third paper in a series of three papers. The first two papers in the series are called "Artinian Gorenstein algebras with linear resolutions", (arXiv:1306.2523) and "The explicit minimal resolution constructed from a Macaulay inverse system". In the present paper, we give the explicit minimal resolution of an Artinian Gorenstein algebra with a linear resolution. This minimal resolution is given in a polynomial manner in terms of the coefficients of the Macaulay inverse system for the Gorenstein algebra. Let k be a field, A a standard-graded Artinian Gorenstein k-algebra, S the standard-graded polynomial ring Sym(A_1), I the kernel of the natural surjection from S to A, d the vector space dimension of A_1, and n the least index with I_n not equal to 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. Our description of B depends on a fixed decomposition of A_1 of the form k x_1\oplus V_0, for some non-zero element x_1 and some d-1 dimensional subspace V_0 of A_1. Much information about B is already contained in the complex Bbar=B/x_1B, 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 Bbar. The skeleton Bbar 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 Bbar into Schur and Weyl modules lifts to a decomposition of B; furthermore, B inherits the natural self-duality of Bbar. As an application we observe that every non-zero element of A_1 is a weak Lefschetz element for A.