The explicit minimal resolution constructed from a Macaulay inverse system

JOURNAL OF ALGEBRA

卷:440

Article

El Khoury, Sabine¹  Kustin, Andrew R.² 

[1] Amer Univ Beirut, Dept Math, Beirut 110236, Lebanon

[2] Univ S Carolina, Dept Math, Columbia, SC 29208 USA

关键词: Artinian rings;    Buchsbaum-Eisenbud ideals;    Build resolution directly from inverse system;    Gorenstein ririgs;    Linear presentation;    Linear resolution;    Macaulay inverse system;    Pfaffians;    Resolutions;   

DOI  :  10.1016/j.jalgebra.2015.04.044

来源: Elsevier

【 摘 要 】

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(center dot)(k) (A(1)) modules, is Gorenstein-linear. Fix a basis x, y, z for the k-vector space A(1). If G is Gorenstein linear, then the socle degree of A is necessarily even, and, if n is the least index with dim(k) A(n) less than dim(k) Sym(n)(k)(A(1)), then the socle degree of A is 2n - 2. Let Phi = Sigma alpha(m)m*, as m roams over the monomials in x, y, z of degree 2n-2, with alpha(m) is an element of k, be an arbitrary homogeneous element of degree 2n-2 in the divided power module D-center dot(k) (A(1)*). The annihilator of Phi (denoted ann Phi) is the ideal of elements f in Sym(center dot)(k) (A(1)) with f(Phi) = 0. The element Phi of D-center dot(k) (A(1)*) is the Macaulay inverse system for the ring Sym(center dot)(k) (A(1)) / ann Phi, which is necessarily Gorenstein and Artinian. Consider the matrix (alpha(mm')), as m and m' roam over the monomials in x, y, z of degree n - 1. The ring Sym(center dot)(k) (A(1)) / ann Phi has a Gorenstein-linear resolution if and only if det(alpha(mm')) not equal 0. If det(alpha(mm')) not equal 0, then we give explicit formulas for the minimal homogeneous resolution of Sym(center dot)(k) (A(1)) / ann Phi in terms the alpha(m)'s and x, y, z. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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