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?



Is the pushforward of a locally free sheaf by an open immersion coherent?
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


