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?
1 Answer
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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.
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2$\begingroup$ alternatively, the result holds simply because $f$ is affine, since it is finite (you don't actually need the semi-continuity theorem). $\endgroup$ Commented Oct 16, 2011 at 7:31