The explicit minimal resolution constructed from a Macaulay inverse system

Abstract

Let A be a standard-graded Artinian Gorenstein algebra of embedding codimension three over a field k. In the generic case, the minimal homogeneous resolution, G, of A, by free Sym•k(A1) modules, is Gorenstein-linear. Fix a basis x, y, z for the k-vector space A1. If G is Gorenstein linear, then the socle degree of A is necessarily even, and, if n is the least index with dimk⁡An less than dimk⁡Symnk(A1), then the socle degree of A is 2n−2. LetΦ=∑αmm⁎, as m roams over the monomials in x, y, z of degree 2n−2, with αm∈k, be an arbitrary homogeneous element of degree 2n−2 in the divided power module D•k(A1⁎). The annihilator of Φ (denoted annΦ) is the ideal of elements f in Sym•k(A1) with f(Φ)=0. The element Φ of D•k(A1⁎) is the Macaulay inverse system for the ring Sym•k(A1)/annΦ, which is necessarily Gorenstein and Artinian. Consider the matrix (αmm′), as m and m′ roam over the monomials in x, y, z of degree n−1. The ring Sym•k(A1)/annΦ has a Gorenstein-linear resolution if and only if det⁡(αmm′)≠0. If det⁡(αmm′)≠0, then we give explicit formulas for the minimal homogeneous resolution of Sym•k(A1)/annΦ in terms the αm's and x, y, z.