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[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[k]]> </alternatives>. Assume that <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91hgr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[depth (R/ a)≥ 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, …, 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[char (k)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_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[in- 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≥ 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≤ 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[depth (R/ a)≥ 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_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[in- 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[k]]> </alternatives> is.