The structure of Gorenstein-linear resolutions of Artinian algebras

TL;DR

This paper explicitly constructs the minimal Gorenstein-linear resolution of Artinian Gorenstein algebras with linear resolutions, revealing a structure determined solely by the algebra's dimension and a fixed decomposition.

Contribution

It provides a polynomial description of the minimal resolution based on the Macaulay inverse system, extending previous work and clarifying the resolution's structure.

Findings

The skeleton of the resolution is fully determined by (d,n)

The resolution inherits self-duality and decomposes into Schur and Weyl modules

Every non-zero element of A_1 acts as a weak Lefschetz element

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.