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Apologies if this is a standard question for algebaric geometry colleagues: Suppose I have a variety, what is the ring Ext(1,1) of self-extensions of the unit object (trivial sheaf) in the categoy of sheaves over the variety. Same question for the category of equivariant sheaves under some action? Is it connected to the cohomology of the variety?

Thanks for any hint or reference, Simon

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    $\begingroup$ What kind of sheaves do you have in mind? What's the "trivial" sheaf? $\endgroup$
    – Angelo
    Commented Jun 16, 2020 at 15:31
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    $\begingroup$ Whatever kind of sheaf theory you have in mind, the expression Ext(1,1) is almost tautologically the cohomology of the variety (with "trivial" coefficients). So for coherent sheaves you get coherent cohomology (of the structure sheaf), for etale l-adic sheaves you get etale cohomology, for D modules you get de rham cohomology, ditto for equivariant versions... $\endgroup$
    – Sam Gunningham
    Commented Jun 16, 2020 at 22:17
  • $\begingroup$ That was fast and exactly what I wanted to know. I appreachiate it a lot, thank you! $\endgroup$
    – Simon Lentner
    Commented Jun 16, 2020 at 23:59

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