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.
- 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.
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?
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?
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.
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 $V$.
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?