Descend finite etale algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:48:37Z http://mathoverflow.net/feeds/question/9158 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9158/descend-finite-etale-algebras Descend finite etale algebras shenghao 2009-12-17T06:47:10Z 2009-12-18T21:00:58Z <p>Let $\pi:X\to\mathcal X$ be a presentation of an Artin stack $\mathcal X$ of finite type over a field $k,$ and let $f:Y\to X$ be a finite \'etale covering. Does there exist a finite \'etale covering $Y'\to X$ factoring through $Y,$ such that $Y'$ can be given descent structure, i.e. there exists an isomorphism <code>$pr_1^*Y'\cong pr_2^*Y'$</code> over <code>$X\times_{\mathcal X}X$</code> satisfying cocycle condition, so that $Y'\to X$ descend to a finite \'etale covering $\mathcal Y\to\mathcal X$?</p> http://mathoverflow.net/questions/9158/descend-finite-etale-algebras/9313#9313 Answer by t3suji for Descend finite etale algebras t3suji 2009-12-18T20:58:23Z 2009-12-18T20:58:23Z <p>I don't think so (finite etale covers cannot be localized in smooth topology in the sense that you describe). Say, $\mathcal{X}$ is a point, and $X$ is a smooth variety with non-trivial fundamental group, say, an elliptic curve (or $\mathbb{A}^1-{0}$). Then $\pi$ is a presentation. Let $f:Y\to X$ be a non-trivial finite etale cover, say, the cover of the elliptic curve by an isogeneous elliptic curve. Then your question becomes: `is there a trivial (i.e., lifted from $\mathcal{X}$) cover $Y'$ of $X$ with a map to $Y$? This is of course not true. </p>