A matrix of linear forms which is annihilated by a vector of indeterminates

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• 作者：Andrew R. Kustin^a ; ^{kustin@math.sc.edu" class="auth_mail" title="E-mail the corresponding author} ; Claudia Polini^b ; ^{cpolini@nd.edu" class="auth_mail" title="E-mail the corresponding author} ; Bernd Ulrich^c ; ^{ulrich@math.purdue.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 13D02 ; secondary ; 13C40 ; 13A30
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：469
• 期：Complete
• 页码：120-187
• 全文大小：884 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si1.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=da33e040a2a67cab0d86a0551e241aaa" title="Click to view the MathML source">R=k[T₁,…,T_f]class="mathContainer hidden">class="mathCode">$R=k[T_1,\dots ,T_{\mathfrak{f}}]$ be a standard graded polynomial ring over the field k   and Ψ be an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si17.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=d837ebf25b79345f1f0758fe51975dcf" title="Click to view the MathML source">f×gclass="mathContainer hidden">class="mathCode">$\mathfrak{f}\times \mathfrak{g}$ matrix of linear forms from R  , where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si3.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=9f6d19d415df51087d17bbd6fde17af4" title="Click to view the MathML source">1≤g<fclass="mathContainer hidden">class="mathCode">$1\le \mathfrak{g}<\mathfrak{f}$. Assume class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si4.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=bd7f97c23acbb889517dcc166518dcdf">class="imgLazyJSB inlineImage" height="17" width="109" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302708-si4.gif">class="mathContainer hidden">class="mathCode">$\left[\begin{array}{ccc}\hfill T_1\hfill & \hfill \cdots \hfill & \hfill T_{\mathfrak{f}}\hfill \end{array}\right]\mathrm{\Psi }$ is 0 and that class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si21.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=df71cd7ce3d4e8236e6d60803c8d81e4">class="imgLazyJSB inlineImage" height="17" width="82" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302708-si21.gif">class="mathContainer hidden">class="mathCode">$\mathrm{grade}{I}_{\mathfrak{g}}(\mathrm{\Psi })$ is exactly one short of the maximum possible grade. We resolve class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si6.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=08fee340632b4bd4f0c63d7ab5435dd2">class="imgLazyJSB inlineImage" height="20" width="95" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302708-si6.gif">class="mathContainer hidden">class="mathCode">$\bar{R}=R/I_{\mathfrak{g}}(\mathrm{\Psi })$, prove that class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si151.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=55cf7177acd17daabddc0907f1fa708b">class="imgLazyJSB inlineImage" height="14" width="15" 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-S0021869316302708-si151.gif">class="mathContainer hidden">class="mathCode">$\bar{R}$ has a class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si186.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=a42d4c1f99ce9716017e4934ce635a44" title="Click to view the MathML source">gclass="mathContainer hidden">class="mathCode">$\mathfrak{g}$-linear resolution, record explicit formulas for the h  -vector and multiplicity of class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si151.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=55cf7177acd17daabddc0907f1fa708b">class="imgLazyJSB inlineImage" height="14" width="15" 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-S0021869316302708-si151.gif">class="mathContainer hidden">class="mathCode">$\bar{R}$, and prove that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si9.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=767bee6afcef3d4dcbaf6d992b896bd2" title="Click to view the MathML source">f−gclass="mathContainer hidden">class="mathCode">$\mathfrak{f}-\mathfrak{g}$ is even, then the ideal class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si10.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=a0c5f1cf3205f50b662e9d4d73bbaa62" title="Click to view the MathML source">I_g(Ψ)class="mathContainer hidden">class="mathCode">$I_{\mathfrak{g}}(\mathrm{\Psi })$ is unmixed. Furthermore, if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si9.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=767bee6afcef3d4dcbaf6d992b896bd2" title="Click to view the MathML source">f−gclass="mathContainer hidden">class="mathCode">$\mathfrak{f}-\mathfrak{g}$ is odd, then we identify an explicit generating set for the unmixed part, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si11.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=ca7b776a8eef666ba672c758d7de8245" title="Click to view the MathML source">I_g(Ψ)^unmclass="mathContainer hidden">class="mathCode">$I_{\mathfrak{g}}{(\mathrm{\Psi })}^{\text{<font color="red">unm</font>}}$, of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si10.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=a0c5f1cf3205f50b662e9d4d73bbaa62" title="Click to view the MathML source">I_g(Ψ)class="mathContainer hidden">class="mathCode">$I_{\mathfrak{g}}(\mathrm{\Psi })$, resolve class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si13.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=c922511c33b1157a15298174313ccbbb" title="Click to view the MathML source">R/I_g(Ψ)^unmclass="mathContainer hidden">class="mathCode">$R/I_{\mathfrak{g}}{(\mathrm{\Psi })}^{\text{<font color="red">unm</font>}}$, and record explicit formulas for the h  -vector of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si13.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=c922511c33b1157a15298174313ccbbb" title="Click to view the MathML source">R/I_g(Ψ)^unmclass="mathContainer hidden">class="mathCode">$R/I_{\mathfrak{g}}{(\mathrm{\Psi })}^{\text{<font color="red">unm</font>}}$. (The rings class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si14.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=d4edb59f81e78f7be3171bfe270d7f12" title="Click to view the MathML source">R/I_g(Ψ)class="mathContainer hidden">class="mathCode">$R/I_{\mathfrak{g}}(\mathrm{\Psi })$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302708&_mathId=si13.gif&_user=111111111&_pii=S0021869316302708&_rdoc=1&_issn=00218693&md5=c922511c33b1157a15298174313ccbbb" title="Click to view the MathML source">R/I_g(Ψ)^unmclass="mathContainer hidden">class="mathCode">$R/I_{\mathfrak{g}}{(\mathrm{\Psi })}^{\text{<font color="red">unm</font>}}$ automatically have the same multiplicity.) These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals.