User homalor - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:02:30Z http://mathoverflow.net/feeds/user/16769 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71410/is-the-following-morphism-etale Is the following morphism etale Homalor 2011-07-27T15:03:16Z 2011-07-27T16:34:04Z <p>Let $Y$ be a reduced noetherian $1$-dimensional scheme such that the normalization morphism $f:X \longrightarrow Y$ is finite. Let $g:Y\longrightarrow Z$ be a finite flat morphism, where $Z$ is a connected (1-dimensional) Dedekind scheme.</p> <p>Suppose that the morphism $g\circ f$ from $X$ to $Z$ is etale.</p> <p><strong>Question.</strong> Is the morphism $g:Y\longrightarrow Z$ etale?</p> <p><strong>Remark.</strong> The hypothesis on the dimension is not necessary probably.</p> <p><strong>Remark.</strong> One may assume $Y$ to be integral.</p> <p><strong>Remark.</strong> To assure that $f$ is finite one may suppose that $Y$ is excellent.</p> <p>Interesting cases one may consider are $Z\subset \mathrm{Spec} \mathbf{Z}$ or $Z\subset \mathbf{P}^1_k$ non-empty open.</p>