Let $f:X\rightarrow Y$ be a birational morphism between smooth varieties and $F$ a torsion free sheaf. Is $f_{\ast} F$ torsion free? If not, are there conditions on either $F$, $f$, $X$ or $Y$ that could ensure the torsion freeness of $F$? For example if $f$ is surj. and $F$ is the canonical bundle on $X$ a theorem of Kollar ensure the torsion freeness of $f_*(\omega_X)$.
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
4
2
|
||||||
|
|
6
|
Actually much more is true. In fact there is the following result, whose proof can be found in [Grothendieck - Dieudonné: EGA 1 (Elements de Géométrie Algebrique), Proposition 8.4.5 page 351]. Proposition. Let $X$, $Y$ be two integral schemes and $f \colon X \to Y$ be a dominant morphism. Then for any torsion-free $\mathcal{O}_X$-module $\mathcal{F}$, the push-forward Kollar's result is much deeper since he proves the torsion-freeness of the higher direct images $R^if_* \omega_X$. |
|||
|
You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
|
5
|
As already mentioned by Donu and Francesco, you don't need such big guns as Kollár's theorem. Also, the Proposition Francesco cites might seem more serious than it is by virtue of being in EGA... The point is this: For an open set $V\subseteq Y$, the module $f_*\mathscr F(V)$ is the same as the module $\mathscr F(f^{-1}V)$. Being torsion on an integral scheme is equivalent to having a non-empty support that is strictly smaller than the ambient scheme. If $f$ is surjective this already gives you what you want, and if it is only dominant you need to think a little more to see that the statement is true. |
|||
|
