Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:06:38Z http://mathoverflow.net/feeds/question/118260 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118260/do-regular-noetherian-schemes-of-dimension-one-only-have-finitely-many-etale-cove Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree Masse 2013-01-07T10:00:29Z 2013-01-07T10:00:29Z <p>Let $X$ be a regular noetherian scheme of dimension one. Let $d$ be an integer.</p> <p><strong>Question.</strong> Are there only finitely many finite etale morphisms $Y\to X$ of degree $d$?</p> <p>I want to exclude finite etale morphisms obtained from the base field (if there is any), i.e., if $X$ is of finite type over a field $k$, I don't want to consider the finite etale covers $X_K\to X$ obtained from finite separable field extensions $k\subset K$.</p> <p>Even if we exclude these finite etale covers, the answer is negative when $X=\mathbf{A}^1_{\mathbf F_p}$. </p> <p>Then again, the answer is positive if $X$ is</p> <ol> <li>an open subscheme of Spec $O_K$, with $O_K$ the ring of integers of a number field;</li> <li>a (not necesarily compact) algebraic curve over an algebraically closed field of characteristic zero.</li> <li>An algebraic curve over a field $k$ of characteristic zero (if we exclude the finite etale covers coming from finite extensions $k\subset K$).</li> </ol> <p>In general the answer is negative as shown above. Which condition on $X$ can we impose to obtain a positive answer.</p> <p>I was thinking about $X$ has to be of characteristic zero. Does this suffice?</p>