Is the pushforward of a locally free sheaf by an open immersion coherent? Let $X$ be a quasi-projective variety, $Y$ a projective variety, and  $f:X \rightarrow Y$ an open immersion. If $\mathcal{F}$ is a locally free coherent sheaf, what can be said about $f_\ast \mathcal{F}$? Is it coherent? Is it torsion free? Is it reflexive?
 A: About your new question:
Let $Y$ be a projective variety and let $X\subset Y$ be an open subset with complement the closed subset $S:=Y\setminus X$. Call $f:X\hookrightarrow Y$ the inclusion.
Let $\mathcal F$ be an algebraic coherent sheaf without torsion on  $X$.
Theorem (Serre-Grothendieck) Suppose that $Y$ is normal and that $S$ has codimension $\geq 2$. Then the sheaf $f_\ast \mathcal F$ is coherent.
Serre, Prolongement de faisceaux analytiques cohérents, Ann.Inst.Fourier 16 (1966), 363-374
A: Dear Yemon,
a)The sheaf  $f_\ast \mathcal{F}$ is not coherent in general since its stalk will not be finitely generated over the local ring of a point of $Y\setminus X$. For example take $P$  a point of $\mathbb P^1=Y$ and $X= \mathbb P^1 \setminus P=\mathbb A^1$. Then for $\mathcal F =\mathcal O_X$, you get $(f_\ast \mathcal{F})_P= Rat(Y)$
b) The direct image $f_\ast \mathcal{F}$  will be torsion free because an inductive limit of torsion free modules over a domain is torsion free ( I assume that variety means in particular integral scheme.)
c) I'm not sure reflexive is a reasonable concept for a non-coherent sheaf.
A: By the way, assuming by varieties, you mean irreducible varieties, then for the second question, the answer is yes.  
For the torsion free-ness, suppose that $r \in H^0(U, O_X)$ kills some non-zero element $z \in H^0(U, f_* \mathcal{F}) = H^0(U \cap X, \mathcal{F})$.  By restriction, $r$ is a non-zero element of $H^0(X \cap U, \mathcal{O}_Z)$.  We still have $rz = 0$ even in this setting, and so by restricting to an affine cover of $X$, it still happens.  This will contradict the torsion-freeness (and thus in particular the locally-freeness) of $\mathcal{F}$.
