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)).

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

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

--
David A. Madore
,
http://www.madore.org/~david/ )