The structure of Gorenstein-linear resolutions of Artinian algebras

详细信息    查看全文

• 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 9316000508&_mathId=si1.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=b84008577aed921c0125a43b6bbf5075">9316000508-si1.gif">${\mathrm{Sym}}_{•}^{\mathit{k}}{A}_1$, I the kernel of the natural map 9316000508-fx001.gif">, d   the vector space dimension 9316000508&_mathId=si17.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=216c51cb6a2d1ab9b898edcaaede6f47" title="Click to view the MathML source">dim_k⁡A₁${\dim }_{\mathit{k}}{A}_1$, and n   the least index with 9316000508&_mathId=si18.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=a258a0e147ee8804ad159f8c3fbad170" title="Click to view the MathML source">I_n≠0$I_n\ne 0$. Assume that 9316000508&_mathId=si4.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=663cdd3645c114b954b7af8cc30aea51" title="Click to view the MathML source">3≤d$3\le d$ and 9316000508&_mathId=si5.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=653bfea67c0fabeb199ec13812ebd7aa" title="Click to view the MathML source">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 9316000508&_mathId=si6.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=76c2f416c948479752dfcfb2f491844a" title="Click to view the MathML source">A₁$A_1$ of the form 9316000508&_mathId=si7.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=d8145264d6acc0c993e10618ec4bb016" title="Click to view the MathML source">kx₁⊕V₀$\mathit{k}{x}_1\oplus V_0$, for some non-zero element 9316000508&_mathId=si8.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=e36df9d3704f96cc8e871ab99c9a90aa" title="Click to view the MathML source">x₁$x_1$ and some 9316000508&_mathId=si9.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=e28e88db37c16c72733daa71a766de07" title="Click to view the MathML source">(d−1)$(d-1)$ dimensional subspace 9316000508&_mathId=si10.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=b576ab5449e4f39be2719dd34d99c3ca" title="Click to view the MathML source">V₀$V_0$ of 9316000508&_mathId=si6.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=76c2f416c948479752dfcfb2f491844a" title="Click to view the MathML source">A₁$A_1$. Much information about B is already contained in the complex 9316000508&_mathId=si11.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=a484cd7cfa90029bc520309cfe01d9bf">9316000508-si11.gif">$\bar{\mathbf{B}}=\mathbf{B}/x_1\mathbf{B}$, 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 9316000508&_mathId=si12.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=12e07350bfbe8ba10cc0c7abd621143f" title="Click to view the MathML source">(d,n)$(d,n)$; no other information about A   is used in the construction of 9316000508&_mathId=si13.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=303780f786c38cee8a2ce32661acc6b2">9316000508-si13.gif">$\bar{\mathbf{B}}$.

The skeleton 9316000508&_mathId=si13.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=303780f786c38cee8a2ce32661acc6b2">9316000508-si13.gif">$\bar{\mathbf{B}}$ is the mapping cone of 9316000508&_mathId=si14.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=3c0decc6b09e5200a9ed7fc8f28b31a8" title="Click to view the MathML source">zero:K→L$\mathrm{zero}:\mathbb{K}\to \mathbb{L}$, where 9316000508&_mathId=si15.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=644f1522c6197c13f697dba6749289c7" title="Click to view the MathML source">L$\mathbb{L}$ is a well known resolution of Buchsbaum and Eisenbud; 9316000508&_mathId=si16.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=0e9d9fed6c57177363e9c926c69f3fa0" title="Click to view the MathML source">K$\mathbb{K}$ is the dual of 9316000508&_mathId=si15.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=644f1522c6197c13f697dba6749289c7" title="Click to view the MathML source">L$\mathbb{L}$; and 9316000508&_mathId=si15.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=644f1522c6197c13f697dba6749289c7" title="Click to view the MathML source">L$\mathbb{L}$ and 9316000508&_mathId=si16.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=0e9d9fed6c57177363e9c926c69f3fa0" title="Click to view the MathML source">K$\mathbb{K}$ are comprised of Schur and Weyl modules associated to hooks, respectively. The decomposition of 9316000508&_mathId=si13.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=303780f786c38cee8a2ce32661acc6b2">9316000508-si13.gif">$\bar{\mathbf{B}}$ into Schur and Weyl modules lifts to a decomposition of B; furthermore, B inherits the natural self-duality of 9316000508&_mathId=si13.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=303780f786c38cee8a2ce32661acc6b2">9316000508-si13.gif">$\bar{\mathbf{B}}$.

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 9316000508&_mathId=si13.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=303780f786c38cee8a2ce32661acc6b2">9316000508-si13.gif">$\bar{\mathbf{B}}$, 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 9316000508&_mathId=si8.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=e36df9d3704f96cc8e871ab99c9a90aa" title="Click to view the MathML source">x₁$x_1$. As an application we observe that every non-zero element of 9316000508&_mathId=si6.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=76c2f416c948479752dfcfb2f491844a" title="Click to view the MathML source">A₁$A_1$ is a weak Lefschetz element for A.