When is the pushforward of a vector bundle still a vector bundle? Let $X$ and $Y$ be varieties. Let $E$ be a locally free sheaf over $X$. Let $f: X \to Y$. Is there some nice criteria which ensures that $f_\ast E$ is still locally free? Sorry, if this is a very standard question.
 A: Check out Grauert's Theorem, in Hartshorne III.12.
A: Under reasonable hypotheses on $X$, $Y$ and $f$, the answer is that $f_*E$ is locally free if and only if $\dim H^0(X_y, \, E_y)$ is a constant function, where
$$X_y:=f^{-1}(y), \quad E_y:=E|_{X_y}.$$
More precisely, there is the following result, whose proof can be found in [Mumford, Abelian Varieties, Chapter II]:

Theorem (Base Change). Let $f \colon X \to Y$ be a proper morphism of Noetherian schemes, with $Y$ reduced and connected, and $E$ a coherent sheaf on $X$, flat over $Y$. Then for all integers $p \geq 0$  the following conditions are equivalent:
$\boldsymbol{(i)}$ $y \to \dim H^p(X_y, E_y)$ is a constant function;
$\boldsymbol{(ii)}$ $F:=R^pf_*E$ is a locally free sheaf on $Y$ and, for all $y \in Y$, the natural map
$$F \otimes_{\mathcal{O}_Y} k(y) \to H^p(X_y, E_y)$$
is an isomorphism.

For instance, if $f$ is a finite map and $E$ is a line bundle we obtain that $f_*E$ is a vector bundle with $\operatorname{rank} f_*E=\deg f$, whereas $R^p f_*E=0$ for $p >0$.
