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 lass="boldFont">k be a field, A a standard-graded Artinian Gorenstein lass="boldFont">k-algebra, S   the standard-graded polynomial ring lsi1" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si1.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=b84008577aed921c0125a43b6bbf5075">lass="imgLazyJSB inlineImage" height="19" width="62" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316000508-si1.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">l">Sym•ld-italic">kA1, I the kernel of the natural map lass="imgLazyJSB inlineImage" height="12" width="71" alt="Full-size image (1 K)" title="Full-size image (1 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316000508-fx001.gif">, d   the vector space dimension lsi17" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si17.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=216c51cb6a2d1ab9b898edcaaede6f47" title="Click to view the MathML source">dim_k⁡A₁lass="mathContainer hidden">lass="mathCode">low="scroll">l">dimld-italic">k⁡A1, and n   the least index with lsi18" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si18.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=a258a0e147ee8804ad159f8c3fbad170" title="Click to view the MathML source">I_n≠0lass="mathContainer hidden">lass="mathCode">low="scroll">In≠0. Assume that lsi4" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si4.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=663cdd3645c114b954b7af8cc30aea51" title="Click to view the MathML source">3≤dlass="mathContainer hidden">lass="mathCode">low="scroll">3≤d and lsi5" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si5.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=653bfea67c0fabeb199ec13812ebd7aa" title="Click to view the MathML source">2≤nlass="mathContainer hidden">lass="mathCode">low="scroll">2≤n. In this paper, we give the structure of the minimal homogeneous resolution lass="boldFont">B of A by free S-modules, provided lass="boldFont">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 lass="boldFont">B depends on a fixed, but arbitrary, decomposition of lsi6" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si6.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=76c2f416c948479752dfcfb2f491844a" title="Click to view the MathML source">A₁lass="mathContainer hidden">lass="mathCode">low="scroll">A1 of the form lsi7" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si7.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=d8145264d6acc0c993e10618ec4bb016" title="Click to view the MathML source">kx₁⊕V₀lass="mathContainer hidden">lass="mathCode">low="scroll">ld-italic">kx1⊕V0, for some non-zero element lsi8" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si8.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=e36df9d3704f96cc8e871ab99c9a90aa" title="Click to view the MathML source">x₁lass="mathContainer hidden">lass="mathCode">low="scroll">x1 and some lsi9" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si9.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=e28e88db37c16c72733daa71a766de07" title="Click to view the MathML source">(d−1)lass="mathContainer hidden">lass="mathCode">low="scroll">lse">(d−1lse">) dimensional subspace lsi10" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si10.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=b576ab5449e4f39be2719dd34d99c3ca" title="Click to view the MathML source">V₀lass="mathContainer hidden">lass="mathCode">low="scroll">V0 of lsi6" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si6.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=76c2f416c948479752dfcfb2f491844a" title="Click to view the MathML source">A₁lass="mathContainer hidden">lass="mathCode">low="scroll">A1. Much information about lass="boldFont">B is already contained in the complex lsi11" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si11.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=a484cd7cfa90029bc520309cfe01d9bf">lass="imgLazyJSB inlineImage" height="18" width="87" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316000508-si11.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">ld">B‾=ld">Blse">/x1ld">B, which we call the skeleton of lass="boldFont">B. One striking feature of lass="boldFont">B is the fact that the skeleton of lass="boldFont">B is completely determined by the data lsi12" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si12.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=12e07350bfbe8ba10cc0c7abd621143f" title="Click to view the MathML source">(d,n)lass="mathContainer hidden">lass="mathCode">low="scroll">lse">(d,nlse">); no other information about A   is used in the construction of lsi13" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si13.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=303780f786c38cee8a2ce32661acc6b2">lass="imgLazyJSB inlineImage" height="14" width="16" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316000508-si13.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">ld">B‾.

The skeleton lsi13" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si13.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=303780f786c38cee8a2ce32661acc6b2">lass="imgLazyJSB inlineImage" height="14" width="16" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316000508-si13.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">ld">B‾ is the mapping cone of lsi14" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si14.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=3c0decc6b09e5200a9ed7fc8f28b31a8" title="Click to view the MathML source">zero:K→Llass="mathContainer hidden">lass="mathCode">low="scroll">l">zero:le-struck">Klse">→le-struck">L, where lsi15" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si15.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=644f1522c6197c13f697dba6749289c7" title="Click to view the MathML source">Llass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">L is a well known resolution of Buchsbaum and Eisenbud; lsi16" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si16.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=0e9d9fed6c57177363e9c926c69f3fa0" title="Click to view the MathML source">Klass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">K is the dual of lsi15" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si15.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=644f1522c6197c13f697dba6749289c7" title="Click to view the MathML source">Llass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">L; and lsi15" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si15.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=644f1522c6197c13f697dba6749289c7" title="Click to view the MathML source">Llass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">L and lsi16" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si16.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=0e9d9fed6c57177363e9c926c69f3fa0" title="Click to view the MathML source">Klass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">K are comprised of Schur and Weyl modules associated to hooks, respectively. The decomposition of lsi13" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si13.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=303780f786c38cee8a2ce32661acc6b2">lass="imgLazyJSB inlineImage" height="14" width="16" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316000508-si13.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">ld">B‾ into Schur and Weyl modules lifts to a decomposition of lass="boldFont">B; furthermore, lass="boldFont">B inherits the natural self-duality of lsi13" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si13.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=303780f786c38cee8a2ce32661acc6b2">lass="imgLazyJSB inlineImage" height="14" width="16" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316000508-si13.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">ld">B‾.

The differentials of lass="boldFont">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 lsi13" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si13.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=303780f786c38cee8a2ce32661acc6b2">lass="imgLazyJSB inlineImage" height="14" width="16" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316000508-si13.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">ld">B‾, the description of the differentials of lass="boldFont">B amounts to giving a minimal generating set of I  , and, for the interior differentials, giving the coefficients of lsi8" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si8.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=e36df9d3704f96cc8e871ab99c9a90aa" title="Click to view the MathML source">x₁lass="mathContainer hidden">lass="mathCode">low="scroll">x1. As an application we observe that every non-zero element of lsi6" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316000508&_mathId=si6.gif&_user=111111111&_pii=S0021869316000508&_rdoc=1&_issn=00218693&md5=76c2f416c948479752dfcfb2f491844a" title="Click to view the MathML source">A₁lass="mathContainer hidden">lass="mathCode">low="scroll">A1 is a weak Lefschetz element for A.