User george p. - MathOverflow

Why does a group action on a scheme induce a group action on cohomology?

2011-11-01T09:41:23Z

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)?

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.

Comment by George P.

2011-11-01T13:22:08Z

I just came across the book "Etale Cohomology Theory" by Lei Fu which seems to explain precisely this stuff with the G-sheaves you mentioned. I will take a look at this now.

Comment by George P.

2011-11-01T11:48:34Z

Very good, thanks, I'll think about this! Two more questions: 1. What is a constant group? 2. Do you have a good reference for what you just explained? Perhaps something post-SGA to make it shorter :)