Lifting Etale Morphisms - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:03:27Z http://mathoverflow.net/feeds/question/23624 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23624/lifting-etale-morphisms Lifting Etale Morphisms Charles Staats 2010-05-05T20:37:46Z 2010-05-05T22:02:32Z <p>I am trying to find a reference for the following theorem:</p> <p>Let $R$ be a complete DVR, and let $Y$ be a scheme projective and flat over $R$. Suppose that $X_0 \longrightarrow Y_0$ is a finite etale morphism, where $Y_0$ is the fiber of $Y$ over the unique closed point of Spec $R$. Then there exists a scheme $X$, finite etale over $Y$, together with a closed immersion $X_0 \longrightarrow X$ such that $X_0 = X \times_Y Y_0$.</p> <p>In a special case, this allows us to lift etale morphisms in characteristic $p$ to etale morphisms in characteristic zero, and can thus be important in studying the etale fundamental group of varieties in finite characteristic.</p> <p>Note: While any reference would be appreciated, what I am really interested in is a reference that I can give for attribution purposes. My advisor thinks that this result was proved by Deligne about 40 years ago, but I would like something a little more solid.</p> http://mathoverflow.net/questions/23624/lifting-etale-morphisms/23636#23636 Answer by JS Milne for Lifting Etale Morphisms JS Milne 2010-05-05T22:02:32Z 2010-05-05T22:02:32Z <p>SGA 1, IX, 1.10: Let $Y$ be a scheme proper over a complete local noetherian ring $R$, and let $Y_0$ be the closed fibre of $Y/R$. Then the functor $X\mapsto X_0$ from finite etale coverings of $Y$ to finite etale coverings of $Y_0$ is an equivalence of categories.</p>