The structure of Gorenstein-linear resolutions of Artinian algebras

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• 作者：Sabine El Khoury^a ; ^{se24@aub.edu.lb" class="auth_mail" title="E-mail the corresponding author} ; Andrew R. Kustin^b ; ¹ ; ^{kustin@math.sc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 13D02 ; 13E10 ; 13H10 ; 13A02 ; 20G05
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：453
• 期：Complete
• 页码：492-560
• 全文大小：916 K

文摘

Let k be a field, A a standard-graded Artinian Gorenstein k-algebra, S   the standard-graded polynomial ring ${\mathrm{Sym}}_{•}^{\mathit{k}}{A}_1$, I the kernel of the natural map , d   the vector space dimension dim_k⁡A₁${\dim }_{\mathit{k}}{A}_1$, and n   the least index with I_n≠0$I_n\ne 0$. Assume that 3≤d$3\le d$ and 2≤n$2\le n$. 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$A_1$ of the form e10618ec4bb016" title="Click to view the MathML source">kx₁⊕V₀$\mathit{k}{x}_1\oplus V_0$, for some non-zero element x₁$x_1$ and some (d−1)$(d-1)$ dimensional subspace V₀$V_0$ of A₁$A_1$. Much information about B is already contained in the complex $\bar{\mathbf{B}}=\mathbf{B}/x_1\mathbf{B}$, 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)$(d,n)$; no other information about A   is used in the construction of $\bar{\mathbf{B}}$.

The skeleton $\bar{\mathbf{B}}$ is the mapping cone of zero:K→L$\mathrm{zero}:\mathbb{K}\to \mathbb{L}$, where L$\mathbb{L}$ is a well known resolution of Buchsbaum and Eisenbud; K$\mathbb{K}$ is the dual of L$\mathbb{L}$; and L$\mathbb{L}$ and K$\mathbb{K}$ are comprised of Schur and Weyl modules associated to hooks, respectively. The decomposition of $\bar{\mathbf{B}}$ into Schur and Weyl modules lifts to a decomposition of B; furthermore, B inherits the natural self-duality of $\bar{\mathbf{B}}$.

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 $\bar{\mathbf{B}}$, 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$x_1$. As an application we observe that every non-zero element of A₁$A_1$ is a weak Lefschetz element for A.