کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4584122 | 1630474 | 2015 | 42 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: The explicit minimal resolution constructed from a Macaulay inverse system The explicit minimal resolution constructed from a Macaulay inverse system](/preview/png/4584122.png)
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, GG, of A , by free Sym
• k(A1) modules, is Gorenstein-linear. Fix a basis x, y, z for the k -vector space A1A1. If GG is Gorenstein linear, then the socle degree of A is necessarily even, and, if n is the least index with dimkAndimkAn less than dimkSymnk(A1), then the socle degree of A is 2n−22n−2. LetΦ=∑αmm⁎,Φ=∑αmm⁎, as m roams over the monomials in x, y, z of degree 2n−22n−2, with αm∈kαm∈k, be an arbitrary homogeneous element of degree 2n−22n−2 in the divided power module D
• k(A1⁎). The annihilator of Φ (denoted ann Φ) is the ideal of elements f in Sym
• k(A1) with f(Φ)=0f(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′)(αmm′), as m and m′m′ roam over the monomials in x, y, z of degree n−1n−1. The ring Sym
• k(A1)/annΦ has a Gorenstein-linear resolution if and only if det(αmm′)≠0det(αmm′)≠0. If det(αmm′)≠0det(αmm′)≠0, then we give explicit formulas for the minimal homogeneous resolution of Sym
• k(A1)/annΦ in terms the αmαm's and x, y, z.
Journal: Journal of Algebra - Volume 440, 15 October 2015, Pages 145–186