etale morphism and direct image - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:38:36Z http://mathoverflow.net/feeds/question/78252 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78252/etale-morphism-and-direct-image etale morphism and direct image yuvi 2011-10-16T06:56:20Z 2011-10-16T07:29:02Z <p>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?</p> http://mathoverflow.net/questions/78252/etale-morphism-and-direct-image/78254#78254 Answer by Damian Rössler for etale morphism and direct image Damian Rössler 2011-10-16T07:29:02Z 2011-10-16T07:29:02Z <p>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.</p>