User george p. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:34:14Z http://mathoverflow.net/feeds/user/18935 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79693/why-does-a-group-action-on-a-scheme-induce-a-group-action-on-cohomology Why does a group action on a scheme induce a group action on cohomology? George P. 2011-11-01T09:41:23Z 2011-11-01T16:30:43Z <p>This is probably totally obvious but I have no clue how this is done: Say you have an endomorphism $f:X \rightarrow X$ of schemes. Why (if true, perhaps some additional assumptions are necessary!) do you get for a Zariski/étale/l-adic sheaf $\mathcal{F}$ on $X$ an induced endomorphism on the corresponding cohomology? How is this constructed? Are there conditions, when the induced morphism is an isomorphism (I'm having a Frobenius in mind)?</p> <p>Perhaps the above is too general, so my real question is: Why does a group/monoid action on the Deligne-Lusztig variety induce a group/monoid action on the l-adic cohomology (with compact support) of this variety? In every book I looked at this is just mentioned but not explained.</p> http://mathoverflow.net/questions/79693/why-does-a-group-action-on-a-scheme-induce-a-group-action-on-cohomology/79697#79697 Comment by George P. George P. 2011-11-01T13:22:08Z 2011-11-01T13:22:08Z I just came across the book &quot;Etale Cohomology Theory&quot; by Lei Fu which seems to explain precisely this stuff with the G-sheaves you mentioned. I will take a look at this now. http://mathoverflow.net/questions/79693/why-does-a-group-action-on-a-scheme-induce-a-group-action-on-cohomology/79697#79697 Comment by George P. George P. 2011-11-01T11:48:34Z 2011-11-01T11:48:34Z Very good, thanks, I'll think about this! Two more questions: 1. What is a <i>constant</i> group? 2. Do you have a good reference for what you just explained? Perhaps something post-SGA to make it shorter :)