کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6414351 | 1630461 | 2016 | 69 صفحه PDF | دانلود رایگان |
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 dimkâ¡A1, and n the least index with Inâ 0. Assume that 3â¤d and 2â¤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 kx1âV0, for some non-zero element x1 and some (dâ1) dimensional subspace V0 of A1. Much information about B is already contained in the complex Bâ¾=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 Bâ¾.The skeleton Bâ¾ is the mapping cone of zero:KâL, 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 Bâ¾ into Schur and Weyl modules lifts to a decomposition of B; furthermore, B inherits the natural self-duality of Bâ¾.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 Bâ¾, 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.
Journal: Journal of Algebra - Volume 453, 1 May 2016, Pages 492-560