Cohomological and projective dimensions. (July 2013)

• Record Type: Journal Article
• Authors: Varbaro, Matteo
• Is Part Of: Compositio mathematica. Volume 149:Number 7(2013)
• Page Start: 1203
• Page End: 1210
• Publication Date: 2013-07
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X12000899
• Languages: English
• ISSNs: 0010-437X

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>Let <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91h47" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathfrak{a}$]]></tex-math></alternatives></inline-formula> be a homogeneous ideal of a polynomial ring <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91dhr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$R$]]></tex-math></alternatives></inline-formula> in <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91j8z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$n$]]></tex-math></alternatives></inline-formula> variables over a field <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91j6v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathbb{k}$]]></tex-math></alternatives></inline-formula>. Assume that <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91hgr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathrm{depth} (R/ \mathfrak{a})\geq t$]]></tex-math></alternatives></inline-formula>, where <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91j7d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$t$]]></tex-math></alternatives></inline-formula> is some number in <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91cw9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\{ 0, \ldots, n\} $]]></tex-math></alternatives></inline-formula>. A result of Peskine and Szpiro says that if <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91f00" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathrm{char} (\mathbb{k})\gt 0$]]></tex-math></alternatives></inline-formula>, then the local cohomology modules <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91jkx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${ H}_{\mathfrak{a}}^{i} (M)$]]></tex-math></alternatives></inline-formula> vanish for all <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91gg7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$i\gt n- t$]]></tex-math></alternatives></inline-formula> and all <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91f7v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$R$]]></tex-math></alternatives></inline-formula>-modules <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91cjs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$M$]]></tex-math></alternatives></inline-formula>. In characteristic <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91k02" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$0$]]></tex-math></alternatives></inline-formula>, there are counterexamples to this for all <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91f23" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$t\geq 4$]]></tex-math></alternatives></inline-formula>. On the other hand, when <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91fvs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$t\leq 2$]]></tex-math></alternatives></inline-formula>, by exploiting classical results of Grothendieck, Lichtenbaum, Hartshorne and Ogus it is not difficult to extend the result to any characteristic. In this paper we settle the remaining case; specifically, we show that if <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91dxc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathrm{depth} (R/ \mathfrak{a})\geq 3$]]></tex-math></alternatives></inline-formula>, then the local cohomology modules <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91ckb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${ H}_{\mathfrak{a}}^{i} (M)$]]></tex-math></alternatives></inline-formula> vanish for all <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91k1m" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$i\gt n- 3$]]></tex-math></alternatives></inline-formula> and all <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91g4q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$R$]]></tex-math></alternatives></inline-formula>-modules <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91fbh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$M$]]></tex-math></alternatives></inline-formula>, whatever the characteristic of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91d0g" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathbb{k}$]]></tex-math></alternatives></inline-formula> is.</p> </abstract>