On 17.05.2008 16:15, David Madore wrote:
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?

The subtle point in example III.9.7.1 is that the morphism stems from
the normalization process, so it is affine - and everything is fine
since things can be easily considered under one single affine chart such
that the tool-kit of commutative algebra can be applied. But our
morphism in the original problem is complete or projective. They are not
affine in general.

The direct image is thoroughly examined in terms of the cohomology
groups H^i(X,F) or higher direct images R^i(f_*)(F). This might be a bit
disguised, but basically direct images are considered by cohomological
tools noted before.

In truth, I find it very annoying that this question is not mentioned
in standard texts on algebraic geometry and commutative algebra.

I understand. I would expect some remarks on that, too, together with
examples in chapters on flat complete morphisms.

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 things get a bit more complicated if a fibre is not contained
in a single _affine_ subset. In general this is the case when complete
morphisms are considered.

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.

I agree, but I think the assumption on f to be projective needs to be
thoroughly used at this point to get a decent result. As it is the
properness of f which guarantees that the direct image of a coherent
sheaf is coherent I was wondering where some ample line bundle should
show up in the argument.

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.

Basically your argument can be extended by having a look at the Cech
complex C^i of E and some affine covering of Y. By assumption C^i are
flat A-modules, but C^i is not exact in general. Taking some twist of E
high enough (w.r.t. an ample line bundle L on Y) then C^*(E(n)) is exact
such that it is a flat resolution of finite length of M:=H^0(Y,E(n)).
This implies that the A-module M is flat. By properness the A-module is
finitely generated, hence locally free. (This is the idea of one of the
implications proven in Hartshorne, Thm. III.9.9.)

I need to think further about this...

.... looking forward to seeing a counterexample on this. :-) If I can
find some more time I will try to find a counterexample as well.

--
Best wishes,
J.