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 mg" 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]

mg="si1.gif" overflow="scroll"> R = k [T_{1}, \dots, T_{f}]

be a standard graded polynomial ring over the field k and Ψ be an mg" 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×g

mg="si17.gif" overflow="scroll"> f \times g

matrix of linear forms from R , where mg" 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<f

mg="si3.gif" overflow="scroll"> 1 \leq g < f

. Assume mg" 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">mg 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">

mg height="17" border="0" style="vertical-align:bottom" width="109" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869316302708-si4.gif">

mg="si4.gif" overflow="scroll"> [\begin{matrix} T_{1} & \dots & T_{f} \end{matrix}] Ψ

is 0 and that mg" 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">mg 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">

mg height="17" border="0" style="vertical-align:bottom" width="82" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869316302708-si21.gif">

mg="si21.gif" overflow="scroll"> grade I_{g} (Ψ)

is exactly one short of the maximum possible grade. We resolve mg" 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">mg 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">

mg height="20" border="0" style="vertical-align:bottom" width="95" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869316302708-si6.gif">

mg="si6.gif" overflow="scroll"> \overline{R} = R / I_{g} (Ψ)

, prove that mg" 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">mg 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">

mg height="14" border="0" style="vertical-align:bottom" width="15" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869316302708-si151.gif">

mg="si151.gif" overflow="scroll"> \overline{R}

has a mg" 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">g

mg="si186.gif" overflow="scroll"> g

-linear resolution, record explicit formulas for the h -vector and multiplicity of mg" 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">mg 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">

mg="si151.gif" overflow="scroll"> \overline{R}

, and prove that if mg" 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−g

mg="si9.gif" overflow="scroll"> f - g

is even, then the ideal mg" 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)

mg="si10.gif" overflow="scroll"> I_{g} (Ψ)

is unmixed. Furthermore, if mg" 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−g

mg="si9.gif" overflow="scroll"> f - g

is odd, then we identify an explicit generating set for the unmixed part, mg" 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)^unm

mg="si11.gif" overflow="scroll"> I_{g} {(Ψ)}^{unm}

, of mg" 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)

mg="si10.gif" overflow="scroll"> I_{g} (Ψ)

, resolve mg" 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)^unm

mg="si13.gif" overflow="scroll"> R / I_{g} {(Ψ)}^{unm}

, and record explicit formulas for the h -vector of mg" 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)^unm

mg="si13.gif" overflow="scroll"> R / I_{g} {(Ψ)}^{unm}

. (The rings mg" 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)

mg="si14.gif" overflow="scroll"> R / I_{g} (Ψ)

and mg" 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)^unm

mg="si13.gif" overflow="scroll"> R / I_{g} {(Ψ)}^{unm}

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.

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