Jannick Asmus in litteris
scripsit:
On 16.05.2008 11:15, David Madore wrote:
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. (*)
By Zeus, you're right. I was confused by Hartshorne's wording of
example III.9.7.1, where he writes that some morphism f:Y-X is not
flat, for "if it were, then f_*(O_Y) would be a flat sheaf of
O_X-modules" [note: I inverted X's and Y's w.r.t. Hartshorne's
notation, for consistency with the original poster's notation]. At
the very least, Hartshorne is horribly confusing here, because never
up to that point is the direct image of a sheaf under a flat morphism
mentioned, and suddenly it pops up without explanation. What did he
have in mind?
In truth, I find it very annoying that this question is not mentioned
in standard texts on algebraic geometry and commutative algebra. If
you have the affine case and quasi-coherent sheaves in mind, the
direct image of a quasi-coherent sheaf by a morphism Spec(B)-Spec(A)
corresponds to viewing a B-module as the induced A-module; so it's
pretty natural to think that if a morphism is flat then direct image
by it will take flat sheaves to flat sheaves (see below). A decent
book on the subject should contain some explanation as to the relation
between "f_*(E) is flat" and "E is f-flat": either my expectations are
too high or there are no decent textbooks on the subject. :-)
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.
I don't think the distinction between "projective" and "proper" is the
issue here, but I may be wrong, of course.
Honestly I doubt that the claim is true as stated although I do not have
any counterexample at hand at the moment.
I also thought this statement was fishy, but I couldn't find a
counterexample, so I was confused as explained above. But this
confusing being cleared away, I agree with you that the statement is
probably wrong without further hypotheses.
However, I can see more clearly the reason why finding a
counterexample is not obvious. Let f:Y-X be a morphism of finite
type between noetherian schemes, and assume X = Spec A is affine
(which we can do, because our problem is local on the target).
Consider a finite covering of Y by affine subschemes V_i = Spec(B_i),
each B_i being an A-algebra (of finite type). If E is a
quasi-coherent sheaf on Y, the restriction E|V_i of E to each V_i is
some (M_i)~ with M_i a B_i-module; the direct image f_*(E|V_i) is
([A]M_i)~ where [A]M_i is M_i seen as an A-module; if we assume that E
and f are flat (or more generally that E is flat over X), then the
[A]M_i are flat, so the f_*(E|V_i) are (this is essentially what
Hartshorne says around II.5 and III.9). Now this does not imply that
f_*(E) is flat as I was too quick to believe, but it is true that
f_*(E) injects in the direct sum sum of the f_*(E|V_i) which are flat
(so f_*(E) is, e.g., torsion free). Now it is not true in general
that a subsheaf (submodule) of a flat sheaf (module) is flat, but it
is true if, say, X is a smooth curve. So my attempts to find a
counterexample with X = A^1 were doomed.
I need to think further about this...
--
David A. Madore
,
http://www.madore.org/~david/ )