Multiple structures with arbitrarily large projective dimension supported on linear subspaces

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• 作者：Craig Huneke^a ; ^{huneke@virginia.edu" class="auth_mail" title="E-mail the corresponding author} ; Paolo Mantero^b ; ^{pmantero@uark.edu" class="auth_mail" title="E-mail the corresponding author} ; Jason McCullough^c ; ^{jmccullough@rider.edu" class="auth_mail" title="E-mail the corresponding author} ; Alexandra Seceleanu^d ; ^{aseceleanu2@math.unl.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 13D05 ; secondary ; 14M06 ; 14M07 ; 13D02
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 February 2016
• 年：2016
• 卷：447
• 期：Complete
• 页码：183-205
• 全文大小：466 K

文摘

Let K   be an algebraically closed field. There has been much interest in characterizing multiple structures in class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004640&_mathId=si1.gif&_user=111111111&_pii=S0021869315004640&_rdoc=1&_issn=00218693&md5=bef091dccbdd4037ef18828f6b97a9d4">class="imgLazyJSB inlineImage" height="17" width="20" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315004640-si1.gif">class="mathContainer hidden">class="mathCode">${\mathbb{P}}_K^n$ defined on a linear subspace of small codimension under additional assumptions (e.g. Cohen–Macaulay). We show that no such finite characterization of multiple structures is possible if one only assumes unmixedness. Specifically, we prove that for any positive integers class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004640&_mathId=si2.gif&_user=111111111&_pii=S0021869315004640&_rdoc=1&_issn=00218693&md5=f21dc593a8c7267f08b0fa384b81e6c5" title="Click to view the MathML source">h,e≥2class="mathContainer hidden">class="mathCode">$h,e\ge 2$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004640&_mathId=si22.gif&_user=111111111&_pii=S0021869315004640&_rdoc=1&_issn=00218693&md5=7e454ca12fb450d480a4ed6778e15fb1" title="Click to view the MathML source">(h,e)≠(2,2)class="mathContainer hidden">class="mathCode">$(h,e)\ne (2,2)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004640&_mathId=si216.gif&_user=111111111&_pii=S0021869315004640&_rdoc=1&_issn=00218693&md5=5e5623875e8c913b54e0a6ea61b1a4c4" title="Click to view the MathML source">p≥5class="mathContainer hidden">class="mathCode">$p\ge 5$ there is a homogeneous ideal I in a polynomial ring over K such that (1) the height of I is h  , (2) the Hilbert–Samuel multiplicity of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004640&_mathId=si5.gif&_user=111111111&_pii=S0021869315004640&_rdoc=1&_issn=00218693&md5=e559e12671ffdfcfa9a7ab4ec5637eef" title="Click to view the MathML source">R/Iclass="mathContainer hidden">class="mathCode">$R/I$ is e  , (3) the projective dimension of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004640&_mathId=si5.gif&_user=111111111&_pii=S0021869315004640&_rdoc=1&_issn=00218693&md5=e559e12671ffdfcfa9a7ab4ec5637eef" title="Click to view the MathML source">R/Iclass="mathContainer hidden">class="mathCode">$R/I$ is at least p and (4) the ideal I   is primary to a linear prime class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004640&_mathId=si6.gif&_user=111111111&_pii=S0021869315004640&_rdoc=1&_issn=00218693&md5=8302b4a6ba41bf491611718f8782394e" title="Click to view the MathML source">(x₁,…,x_h)class="mathContainer hidden">class="mathCode">$(x_1,\dots ,x_h)$. This result is in stark contrast to Manolache's characterization of Cohen–Macaulay multiple structures in codimension 2 and multiplicity at most 4 and also to Engheta's characterization of unmixed ideals of height 2 and multiplicity 2.