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 hmlsrc">he 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">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">hContainer hidden">hCode">h altimg="si1.gif" overflow="scroll">hvariant="normal">Sym•hvariant="bold-italic">kh="0.2em">A1h>, I the kernel of the natural map 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 hmlsrc">hImg" 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⁡A₁hContainer hidden">hCode">h altimg="si17.gif" overflow="scroll">hvariant="normal">dimhvariant="bold-italic">k⁡A1h>, and n   the least index with hmlsrc">hImg" 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≠0hContainer hidden">hCode">h altimg="si18.gif" overflow="scroll">In≠0h>. Assume that hmlsrc">hImg" 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≤dhContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">3≤dh> and hmlsrc">hImg" 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≤nhContainer hidden">hCode">h altimg="si5.gif" overflow="scroll">2≤nh>. 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 hmlsrc">hImg" 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">A₁hContainer hidden">hCode">h altimg="si6.gif" overflow="scroll">A1h> of the form hmlsrc">hImg" 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⊕V₀hContainer hidden">hCode">h altimg="si7.gif" overflow="scroll">hvariant="bold-italic">kx1⊕V0h>, for some non-zero element hmlsrc">hImg" 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">x₁hContainer hidden">hCode">h altimg="si8.gif" overflow="scroll">x1h> and some hmlsrc">hImg" 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)hContainer hidden">hCode">h altimg="si9.gif" overflow="scroll">hy="false">(d−1hy="false">)h> dimensional subspace hmlsrc">hImg" 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">V₀hContainer hidden">hCode">h altimg="si10.gif" overflow="scroll">V0h> of hmlsrc">hImg" 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">A₁hContainer hidden">hCode">h altimg="si6.gif" overflow="scroll">A1h>. Much information about B is already contained in the complex hmlsrc">he 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">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">hContainer hidden">hCode">h altimg="si11.gif" overflow="scroll">hvariant="bold">B‾=hvariant="bold">Bhy="false">/x1hvariant="bold">Bh>, 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 hmlsrc">hImg" 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)hContainer hidden">hCode">h altimg="si12.gif" overflow="scroll">hy="false">(d,nhy="false">)h>; no other information about A   is used in the construction of hmlsrc">he 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">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">hContainer hidden">hCode">h altimg="si13.gif" overflow="scroll">hvariant="bold">B‾h>.

The skeleton hmlsrc">he 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">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">hContainer hidden">hCode">h altimg="si13.gif" overflow="scroll">hvariant="bold">B‾h> is the mapping cone of hmlsrc">hImg" 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→LhContainer hidden">hCode">h altimg="si14.gif" overflow="scroll">hvariant="normal">zero:hvariant="double-struck">Khy="false">→hvariant="double-struck">Lh>, where hmlsrc">hImg" 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">LhContainer hidden">hCode">h altimg="si15.gif" overflow="scroll">hvariant="double-struck">Lh> is a well known resolution of Buchsbaum and Eisenbud; hmlsrc">hImg" 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">KhContainer hidden">hCode">h altimg="si16.gif" overflow="scroll">hvariant="double-struck">Kh> is the dual of hmlsrc">hImg" 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">LhContainer hidden">hCode">h altimg="si15.gif" overflow="scroll">hvariant="double-struck">Lh>; and hmlsrc">hImg" 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">LhContainer hidden">hCode">h altimg="si15.gif" overflow="scroll">hvariant="double-struck">Lh> and hmlsrc">hImg" 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">KhContainer hidden">hCode">h altimg="si16.gif" overflow="scroll">hvariant="double-struck">Kh> are comprised of Schur and Weyl modules associated to hooks, respectively. The decomposition of hmlsrc">he 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">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">hContainer hidden">hCode">h altimg="si13.gif" overflow="scroll">hvariant="bold">B‾h> into Schur and Weyl modules lifts to a decomposition of B; furthermore, B inherits the natural self-duality of hmlsrc">he 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">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">hContainer hidden">hCode">h altimg="si13.gif" overflow="scroll">hvariant="bold">B‾h>.

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 hmlsrc">he 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">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">hContainer hidden">hCode">h altimg="si13.gif" overflow="scroll">hvariant="bold">B‾h>, 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 hmlsrc">hImg" 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">x₁hContainer hidden">hCode">h altimg="si8.gif" overflow="scroll">x1h>. As an application we observe that every non-zero element of hmlsrc">hImg" 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">A₁hContainer hidden">hCode">h altimg="si6.gif" overflow="scroll">A1h> is a weak Lefschetz element for A.