Basic properties of Nisnevich cohomology; \$l'\$-topology? - MathOverflow 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 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 Answer by Thomas Geisser for Basic properties of Nisnevich cohomology; \$l'\$-topology? 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>