Class groups of normal domains over finite fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:55:46Z http://mathoverflow.net/feeds/question/7074 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7074/class-groups-of-normal-domains-over-finite-fields Class groups of normal domains over finite fields Hailong Dao 2009-11-28T19:55:59Z 2010-01-27T09:01:34Z <p>Let R be a local, normal domain of dimension 2. Suppose that R contains a finite field. I am interested in knowing when the class group of R is torsion. In characteristic 0, this is known to be related to R being a rational singularity. </p> <p>Lipman showed that if X is a desingularization of Spec(R), then one has an exact sequence:</p> <p>$0 \to Pic^{0}(X) \to Cl(R) \to H$</p> <p>Here $Pic^{0}(X)$ is the numerically trivial part of the Picard group of $X$, and $H$ is a finite group. Thus the second one is torsion if and only if the first one is. I do not have much understanding of the first group, unfortunately. </p> <p>Does anyone know an answer or reference to this? Does anyone know an example in positive characteristic such that $Cl(R)$ is not torsion? Thanks a lot.</p> http://mathoverflow.net/questions/7074/class-groups-of-normal-domains-over-finite-fields/12038#12038 Answer by Hailong Dao for Class groups of normal domains over finite fields Hailong Dao 2010-01-16T23:13:15Z 2010-01-17T15:53:48Z <p>I recently found some references: Theorem 4.5 of <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6WH2-4CWYY9T-HT&amp;_user=10&amp;_coverDate=06%2F30%2F1975&amp;_alid=1168943407&amp;_rdoc=1&amp;_fmt=high&amp;_orig=search&amp;_cdi=6838&amp;_sort=r&amp;_docanchor=&amp;view=c&amp;_ct=1&amp;_acct=C000050221&amp;_version=1&amp;_urlVersion=0&amp;_userid=10&amp;md5=80ce7ac14ec803da3335ca133a2221db" rel="nofollow"> this paper </a> and Theorem 4 + next Corollary of <a href="http://www.springerlink.com/content/g48k6345752803mr/" rel="nofollow"> this paper </a> which says:</p> <p>If $(R,m,k)$ is a complete normal local domain of dimension $2$ such that $k$ is the algebraic closure of some finite field, then $Cl(R)$ is torsion!</p> <p>It remains open what happens in other situations. </p> http://mathoverflow.net/questions/7074/class-groups-of-normal-domains-over-finite-fields/12073#12073 Answer by Bhargav for Class groups of normal domains over finite fields Bhargav 2010-01-17T04:18:24Z 2010-01-17T13:21:21Z <p>As requested in the comments, here's an example of a local, normal $2$-dimensional domain R in positive characteristic such that $\mathrm{Cl}(R)$ is not torsion: choose an elliptic curve $E \subset \mathbf{P}^2$ over a field $k$ such that $E(k)$ is not torsion, and take R to be the local ring at the origin of the affine cone on $E$ (i.e., $R = k[x,y,z]/(f)_{(x,y,z)}$ where $f$ is a homoegenous cubic defining $E$). This can be done over $k = \overline{\mathbf{F}_p(t)}$.</p> <p><em>Proof</em>: The normality follows from the fact that R is a hypersurface singularity (hence even Gorenstein) and isolated and $2$-dimensional (hence regular in codim 1). Blowing up at the origin defines a map $f:X \to \mathrm{Spec}(R)$. One can then show the following: $X$ is smooth, and $X$ can be identified with the Zariski localisation along the zero section of the total space of the line bundle $L = \mathcal{O}_{\mathbf{P^2}}(-1)|_E$ (these are general facts about cones). By Lipman's theorem, it suffices to show that $\mathrm{Pic}^0(X)$ contains non-torsion elements. As $X$ is fibered over $E$ with a section, the pullback $\mathrm{Pic}^0(E) \to \mathrm{Pic}^0(X)$ is a direct summand. As $\mathrm{Pic}^0(E) \simeq E(k)$ has non-torsion elements by assumption, so does $\mathrm{Pic}^0(X)$.</p> <p>Also, an additional comment: In general, Lipman's theorem tells you that $\mathrm{Cl}(R)$ is torsion if and only if $\mathrm{Pic}^0(X)$ is torsion. Now $\mathrm{Pic}(X) \simeq \lim_n \mathrm{Pic}(X_n)$ where $X_n$ is the $n$-th order thickening of the exceptional fibre $E$. Because we are blowing up a point, the sheaf of ideals $I$ defining $E$ is ample on $E$. The kernel and cokernel of $\mathrm{Pic}(X_n) \to \mathrm{Pic}(X_{n-1})$ are identified with $H^1(E,I|_E^{\otimes n+1})$ and $H^2(E,I|_E^{\otimes n+1})$. As $I|_E$ is ample, it follows that the system "$\lim_n \mathrm{Pic}(X_n)$" is eventually stable. Thus, $\mathrm{Pic}(X) \simeq \mathrm{Pic}(X_n)$ for $n$ sufficiently big. As $X_n$ is a proper variety, it follows that if we are working over a finite field (resp. an algebraic closure of a finite field), then $\mathrm{Pic}^0(X)$ is finite (resp. ind-finite).</p>