Is there a projection formula for motivic étale cohomology? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:01:47Z http://mathoverflow.net/feeds/question/22317 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22317/is-there-a-projection-formula-for-motivic-etale-cohomology Is there a projection formula for motivic étale cohomology? norondion 2010-04-23T08:28:00Z 2010-04-23T20:59:01Z <p>Let $f: X \to Y$ be a finite morphism of varieties with $Y$ smooth. Is there a projection formula for $f$ and $H^{i}_{et}(-,\mathbf{Z}(1)) = H^i(-,\mathbf{G}_m)$?</p> <p>Background: I want to show that for a relative curve $\pi: C \to X$ ($C$, $X$, $\pi$ smooth) with a quasi-section $Y \to C$ closed immersion with $Y \to X$ finite of degree $d$, the kernel of $\pi^*: H^i(X,\mathbf{G}_m) \to H^i(C,\mathbf{G}_m)$ is annihilated by $d$.</p> http://mathoverflow.net/questions/22317/is-there-a-projection-formula-for-motivic-etale-cohomology/22325#22325 Answer by Torsten Ekedahl for Is there a projection formula for motivic étale cohomology? Torsten Ekedahl 2010-04-23T10:24:17Z 2010-04-23T20:59:01Z <p>(As you have also stated the application you have in mind I assume that $X$ has everywhere the same dimension as $Y$ with no embedded components.)</p> <p>We have that <code>$H^i_{et}(X,\mathbf{G}_m) = H^i_{et}(Y,f_\ast\mathbf{G}_m)$</code> as $f$ is finite (this is because $f_\ast$ is exact, see Cor. II:3.5 of Milne: Étale cohomology). If $f$ is also flat we have a norm map $f_\ast\mathbf{G}_m \to \mathbf{G}_m$ whose composite with the inclusion <code>$\mathbf{G}_m \to f_\ast\mathbf{G}_m$</code> is the $d$'th power which gives what you want. In the general case you still have a norm map by first noting that $f$ is flat in codimension $1$ (this comes from the condition I added) and you can take the norm there which then will land in $\mathbf{G}_m$ as $Y$ is normal.</p>