TITLE: Does a locally free sheaf over a product pushforward to a locally free sheaf?
QUESTION [7 upvotes]: Suppose $X$ and $Y$ are two (smooth, affine) algebraic varieties.  Let $\mathcal{F}$ be a locally free coherent sheaf over $X \times Y$, and let $\mathcal{G}$ be the pushforward of $\mathcal{F}$ to $X$.  Is it true that $\mathcal{G}$ is a locally free quasicoherent sheaf?

REPLY [2 votes]: Hi. I don't know how to make a comment; this is on Torsten's example with 
elliptic curve minus zero point. When you take a line bundle corresponding to 
a point (for simplicity not a two torsion point) and add the line bundle corresponding to minus of that point then this rank=2 vector bundle restricted to the elliptic curve minus zero is (even globally) free. 
Indeed, working on the elliptic curve twist with the degree one line bundle L
corresponding to the zero point and take the canonical inclusion of the structure
sheaf on both factors, the quotient must be isomorphic to L squared. 
This is just to understand Bass' theorem on this nice example.