most recent 30 from http://mathoverflow.net 2013-06-18T22:40:51Z http://mathoverflow.net/feeds/question/78697 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78697/basic-properties-of-nisnevich-cohomology-l-topology Mikhail Bondarko 2011-10-20T19:42:19Z 2012-09-30T02:59:38Z

<p>I would like to know more about Nisnevich cohomology (especially, on its properties that could be easily formulated). In particular, I would like to know which of the following statements are true, and what restrictions are needed for them.</p> <ol> <li>The higher Nisnevich cohomology of a smooth (is this necessary?) variety $V$ (over a perfect field $k$) with coefficients in a constant sheaf is zero. </li> </ol> <p>The results of Suslin and Voevodsky reduce this statement to its Zariski counterpart; yet they do not tell much about the case of a non-smooth $V$ (especially, if the characteristic of $k$ is positive, so that one cannot apply cdh-descent). Does there exist an easier reasoning?</p> <ol> <li><p>Assertion 1 seems to imply: for an open embedding $j$ of smooth varieties, the higher direct images $R^ij_*$ of a constant sheaf are zero (for $i>0$). Again, is smoothness necessary here?</p></li> <li><p>Total direct images from derived categories of etale sheaves to those of Nisnevich ones commutes with inverse images with respect to embeddings. It seems easy to check this is stalks; yet I wonder which restrictions are required for this result.</p></li> </ol> <p>Upd. It seems that assertion 1 fails already when $V$ is an irreducible nodal cubic. Hence assertion 2 is wrong for the embedding of a smooth open subvariety of $V$ into $V$.</p> <p>It seems that over characteristic zero field one can fix this by considering cdh-topology instead of the Nisnevich one (and h-topology instead of the etale one; this does not seem to affect the cohomology of smooth varieties with coefficients in constant sheaves). Does anybody know whether a similar scheme could be applied in positive characteristic $p$ to $l$-torsion constant sheaves ($l$ is a prime distinct from $p$) if one considers Gabber's $l'$-topology?</p>

http://mathoverflow.net/questions/78697/basic-properties-of-nisnevich-cohomology-l-topology/108436#108436 Thomas Geisser 2012-09-30T02:59:38Z 2012-09-30T02:59:38Z

<p>For smooth schemes you can use (the same argument as for) a Gersten resolution to show that the cohomology of a constant sheaf agrees with the cohomology of the generic point, and this vanishies as fields have no higher Nisnevich cohomology.</p> <p>I believe you could do the $l$-part in characteristic $p$ with Gabbers $l$-topology. As far as I know, Chisinki and Kelly have some results in this direction.</p>