Cohomological and projective dimensions. (July 2013) - Varbaro, Matteo
Abstract Let <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91h47" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathfrak{a}$]]> </alternatives> be a homogeneous ideal of a polynomial ring <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91dhr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$R$]]> </alternatives> in <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91j8z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$n$]]> </alternatives> variables over a field <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91j6v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathbb{k}$]]> </alternatives>. Assume that <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91hgr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathrm{depth} (R/ \mathfrak{a})\geq t$]]> </alternatives>, where <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91j7d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$t$]]> </alternatives> is some number in <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91cw9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\{ 0, \ldots, n\} $]]> </alternatives>. A result of Peskine and Szpiro says that if <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91f00" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathrm{char} (\mathbb{k})\gt 0$]]> </alternatives>, then the local cohomology modules <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91jkx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${ H}_{\mathfrak{a}}^{i} (M)$]]> </alternatives> vanish for all <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91gg7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$i\gt n- t$]]> </alternatives> and all <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91f7v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$R$]]> </alternatives>-modules <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91cjs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$M$]]> </alternatives>. In characteristic <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91k02" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$0$]]> </alternatives>, there are counterexamples to this for all <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91f23" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$t\geq 4$]]> </alternatives>. On the other hand, when <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91fvs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$t\leq 2$]]> </alternatives>, 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 <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91dxc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathrm{depth} (R/ \mathfrak{a})\geq 3$]]> </alternatives>, then the local cohomology modules <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91ckb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${ H}_{\mathfrak{a}}^{i} (M)$]]> </alternatives> vanish for all <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91k1m" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$i\gt n- 3$]]> </alternatives> and all <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91g4q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$R$]]> </alternatives>-modules <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91fbh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$M$]]> </alternatives>, whatever the characteristic of <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91d0g" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathbb{k}$]]> </alternatives> is.