Let $f:Y\rightarrow X$ be an etale morphism, where $X$ and $Y$ are smooth projective
varieties. Let $V$ be a vector bundle over $Y$. Since $f$ is flat, $V$ is flat over $X$.
Is it true that $f\_\*V$ is flat $\mathcal{O}\_X$-module?



etale morphism and direct image
Since $Y$ and $X$ are proper over the base field, the morphism $f$ is finite (by Zariski's main theorem). Hence by the semi-continuity theorem (see Hartshorne p. 288, Cor. 12.9), $f\_\*V$ is locally free, because $f$ is finite and flat. So the answer is yes.


