Hi,


Is there an example of a proper smooth map of schemes $f:X\to Y$ and a vector bundle $E$ on $X$
such that $f\_\*E$ is not locally free on $Y$?


Thanks



pushforward of locally free sheaf is locally free?
Here is a an example, albeit with $Y$ non reduced:


Let $C$ be a smooth projective curve of genus $g > 0$ over a field $k$ and let $C\_{\epsilon} = C \times\_{k} Spec(k[\epsilon])$ where $k[\epsilon] = k[x]/(x^2)$ is the ring of dual numbers. Let $\mathcal{L}$ be a non-trivial line bundle on $C\_{\epsilon}$ such that its restriction to $C$ is trivial. Such line bundles exist since $g > 0$.


Multiplication by $\epsilon$ gives us an exact sequence of sheaves on $C$
$$ 0 \to \epsilon \mathcal{L} \to \mathcal{L} \to \epsilon\mathcal{L} \to 0 \ .$$


Since $\epsilon \mathcal{L} = \mathcal{L}/\epsilon \mathcal{L}$, it follows from the construction that this sheaf is just $\mathcal{O}\_C$. Since $\mathcal{L}$ is non-trivial the boundary map in the long exact sequence of cohomology from $H^0(C, \mathcal{O}\_C)$ to $H^1(C, \mathcal{O}\_C)$ is non-trivial. Thus $H^0(C, \mathcal{L}) = H^0(C, \mathcal{O}\_C)= k$. 


If we let $X = C\_{\epsilon}$, $Y = Spec(k[\epsilon])$, $f$ the natural map $C\_{\epsilon} \to Spec(k[\epsilon])$, and $E = \mathcal{L}$, it follows from the above that $f\_\*E$ is not locally free.


