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I am looking for a reference for the computation of the sheaf cohomology of a blowup where things are worked out in detail. I'd like to see at least sheaf cohomology of the structure sheaf of the blowup, and if there are general methods for the computation I would be curious about those as well. My searching so far has given me references for singular cohomology of the blowup (Voisin's Complex Geometry and Hodge Theory vol 1 theorem 7.31, or Griffiths-Harris chapter 4 section 6), $\ell$-adic cohomology (SGA5 section VII or SGA7 vol II), and potentially here on MSE though this last one is not detailed enough for me.

I'm primarily interested in blowing up a smooth subvariety of a smooth variety over a field, but more generality would be welcome too.

Hank Scorpio
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  • Sheaf cohomology of which sheaf on the blowup? – Cranium Clamp Mar 29 '23 at 23:59
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    I recall Hartshorne Chapter V having some results about cohomology of blow-ups (in V.3 on monoidal transformations). This is for the case of a surface though (and for the structure sheaves). I don’t think the general case can be obtained from the arguments there. –  Mar 30 '23 at 01:11
  • @CraniumClamp let's start with the structure sheaf. If there's general methods, I'd be interested in those too, I guess. I've added that to the question with an edit. – Hank Scorpio Mar 30 '23 at 02:31
  • @AHappyMathematician I've read that and I agree. Essentially Hartshorne picks the easiest possible case there - the blowup is of a point and the computations go really easily due to the theorem on formal functions, but the method seems to need a fair bit of adjustment for the general case and I don't really see what that adjustment ought to be. – Hank Scorpio Mar 30 '23 at 02:33
  • @HankScorpio Have you tried looking at this Stacks Project post? https://stacks.math.columbia.edu/tag/0FUB –  Mar 31 '23 at 00:24
  • @AHappyMathematician I had not seen that, thanks for the resource! There's some promising stuff in here, but I'm not sure it does everything I need (or maybe I don't quite understand everything there - time to get reading). – Hank Scorpio Apr 01 '23 at 20:19

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