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I hope this is well known, I just could not work it out myself.

Say I have a variety X (smooth and projective over C is my usual setup) with a smooth subvariety Z. Let f: BL_Z(X) --> X be the blowup of X with centre Z and let E be the exceptional.

What is $Rf_* O(kE)$?

Or at least what are its cohomology sheaves? I am mainly interested in $k=\pm 1$.

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    $\begingroup$ Here is one case: $Rf_*O(E)=O_X$. You have a triangle $O_X\to O_X(E)\to O_E$, and the same result is well known for direct image of the first sheaf, and direct image of third should vanish (reduce to cohomology of projective space). It's too late here to write more... $\endgroup$
    – Donu Arapura
    Commented Apr 3, 2015 at 0:48
  • $\begingroup$ ah, yes! I guess the sequence is O --> O(E) --> $O_E(E)$ where the latter can be identified with the relative O(-1), which has no cohomology. $\endgroup$
    – bananastack
    Commented Apr 3, 2015 at 3:18
  • $\begingroup$ You also have $R\pi_*O(-E)=I_Z$. $\endgroup$
    – gsvr
    Commented Apr 4, 2015 at 2:07

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