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Hallo,

in a proof I read an argument which used the following "fact" which was no further explained:

if you have a closed point $x$ on a smooth projective variety over a field $X$ and denote with $k(x)$ the skyskraper sheaf in the point $x$ with the residue class field of $x$ as skyskraper and a coherent sheaf $F$, then one has that the local hypertor

$\tau \omicron r^{-p}(F,k(x))$

is zero for p greater than zero.

Can someone explain why that is the case? I dont know if it is relevant that on $X$ every coherent sheaf has a locally free bounded resolution.

Thanks

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  • $\begingroup$ I can't explain why it's true because it isn't unless $F$ is locally free at $x$. I assume that $Tor_{-p}= Tor_p$. (I didn't downvote, in case you worry about that.) $\endgroup$
    – Donu Arapura
    Commented Aug 13, 2011 at 15:43
  • $\begingroup$ I mean $Tor^{-p}= Tor_p$ $\endgroup$
    – Donu Arapura
    Commented Aug 13, 2011 at 16:10
  • $\begingroup$ ok, thanks. then I'll see if there is some additional hypotheses in the proof which guarantees local freeness or so. $\endgroup$
    – Descartes
    Commented Aug 13, 2011 at 16:48

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