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 241aaa" title="Click to view the MathML source">R=k[T₁,…,T_f]mi>Rmi>=<mi>kmi>[<mi>Tmi>1,…,<mi>Tmi><mi mathvariant="fraktur">fmi>] be a standard graded polynomial ring over the field k   and Ψ be an f×gmi mathvariant="fraktur">fmi>×<mi mathvariant="fraktur">gmi> matrix of linear forms from R  , where 1≤g<fmi mathvariant="fraktur">gmi><<mi mathvariant="fraktur">fmi>. Assume mi>Tmi>1⋯<mi>Tmi><mi mathvariant="fraktur">fmi>]<mi mathvariant="normal">Ψmi> is 0 and that mi mathvariant="normal">grademi><mi>Imi><mi mathvariant="fraktur">gmi>(<mi mathvariant="normal">Ψmi>) is exactly one short of the maximum possible grade. We resolve mi>Rmi>‾=<mi>Rmi>/<mi>Imi><mi mathvariant="fraktur">gmi>(<mi mathvariant="normal">Ψmi>), prove that middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302708-si151.gif">mi>Rmi>‾ has a gmi mathvariant="fraktur">gmi>-linear resolution, record explicit formulas for the h  -vector and multiplicity of middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302708-si151.gif">mi>Rmi>‾, and prove that if f&minus;gmi mathvariant="fraktur">fmi>&minus;<mi mathvariant="fraktur">gmi> is even, then the ideal I_g(Ψ)mi>Imi><mi mathvariant="fraktur">gmi>(<mi mathvariant="normal">Ψmi>) is unmixed. Furthermore, if f&minus;gmi mathvariant="fraktur">fmi>&minus;<mi mathvariant="fraktur">gmi> is odd, then we identify an explicit generating set for the unmixed part, I_g(Ψ)^unmmi>Imi><mi mathvariant="fraktur">gmi>(<mi mathvariant="normal">Ψmi>)unm, of I_g(Ψ)mi>Imi><mi mathvariant="fraktur">gmi>(<mi mathvariant="normal">Ψmi>), resolve R/I_g(Ψ)^unmmi>Rmi>/<mi>Imi><mi mathvariant="fraktur">gmi>(<mi mathvariant="normal">Ψmi>)unm, and record explicit formulas for the h  -vector of R/I_g(Ψ)^unmmi>Rmi>/<mi>Imi><mi mathvariant="fraktur">gmi>(<mi mathvariant="normal">Ψmi>)unm. (The rings R/I_g(Ψ)mi>Rmi>/<mi>Imi><mi mathvariant="fraktur">gmi>(<mi mathvariant="normal">Ψmi>) and R/I_g(Ψ)^unmmi>Rmi>/<mi>Imi><mi mathvariant="fraktur">gmi>(<mi mathvariant="normal">Ψmi>)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.