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direct images of a locally free sheaf
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Zhenya
May 13, 2008, 10:33:54 AM5/13/08
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Hi!
Does anybody know how to prove or disprove the following statement.
Does anybody know how to prove or disprove the following statement.
If f:Y->X is a flat projective morphism, and E - locally free sheaf on
Y, then
1. f_*(E) is locally free
2. R^p f_*(E) are locally free for p > 0
It looks reasonable, at least #1, but I didn't find anything like this
in Hartshorne.
Zhenya
Zhenya
May 13, 2008, 9:32:46 PM5/13/08
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Seems that the statement is wrong, since the people
who use claims like this have additional requirements for E,
e.g. R^p f_*(E) = 0.
who use claims like this have additional requirements for E,
e.g. R^p f_*(E) = 0.
Jannick Asmus
May 14, 2008, 3:32:44 AM5/14/08
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On 13.05.2008 16:33, Zhenya wrote:
> Hi!
> Does anybody know how to prove or disprove the following statement.
>
> If f:Y->X is a flat projective morphism, and E - locally free sheaf on
> Y, then
> 1. f_*(E) is locally free
> 2. R^p f_*(E) are locally free for p > 0
> Hi!
> Does anybody know how to prove or disprove the following statement.
>
> If f:Y->X is a flat projective morphism, and E - locally free sheaf on
> Y, then
> 1. f_*(E) is locally free
> 2. R^p f_*(E) are locally free for p > 0
1.) is true if X is, in addition, integral Noetherian and the function x
-> h^0(x,E)=dim_{k(x)}H^0(Y_x,E_x) is constant on X.
2.) is true if X is Noetherian if E is of the form E(n) for n>>0 where
the twist is taken e.r.t. an f-ample line bundle L on Y.
> It looks reasonable, at least #1, but I didn't find anything like this
> in Hartshorne.
I am quite sure that this is somewhere in Hartshorne's cohomology section.
> Zhenya
Best wishes,
J.
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