On 16.05.2008 11:15, David Madore wrote:
Zhenya in litteris

scripsit:
Does anybody know is this statement right or wrong?

f: Y-X - flat projective morphism
E - locally free sheaf on Y

Then f_* (E) is locally free.

I assume that "locally free" means "locall y free coherent" and that
everything is noetherian.

Then f_*(E) is flat because E is flat and f is (see Hartshorne, *Algebraic
Geometry*, proposition III.9.2(c)).

I can see that the cited proposition gives that _E_ is flat over X. I am
wondering how you deduce that f_*(E) is flat over X with that.
Considering the natural morphism of O_Y-modules f^*f_*(E) - E there
seems more work to be done here. (*)

But f_*(E) is coherent because E is and f is projective
(op. cit. III.8.8(b)).

Note that f_*(E) can be seen to be coherent, too, if f is assumed to be
proper. I think that the strong assumption that f be projective needs to
be invested somehow to get a result.

Now this implies that f_*(E) is locally free (op. cit. III.9.2(e)).

Honestly I doubt that the claim is true as stated although I do not have
any counterexample at hand at the moment.

Note that f_*(E(n)) is locally free for n0 where E(n) is the n-th
twist of E with respect to a f-ample line bundle L on Y if X is integral
(cf. http://groups.google.com/group/sci.m...4a3dfe05992be0, too).
This appears to be the best result which is possible in the light of
Hartshorne, Thm. III.9.9 and the equivalent assertions in its proof.

.... but I am keen on seeing some elaboration on (*).

--
Best wishes,
J.