Reference for Pic(G) and central extensions. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T01:51:52Z http://mathoverflow.net/feeds/question/30265 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30265/reference-for-picg-and-central-extensions Reference for Pic(G) and central extensions. Marty 2010-07-02T03:47:57Z 2010-07-02T20:29:55Z <p>Let $G$ be a connected reductive group over a (perfect, why not) field $F$. Let $m$, $pr_1$, $pr_2$ denote the multiplication, first, and second projection maps from $G \times G$ to $G$.</p> <p>Then I'm pretty sure that I can prove the following fact: if $L$ is a line bundle on $G$, then $m^\ast(L)$ is isomorphic to $pr_1^\ast(L) \cdot pr_2^\ast(L)$. This (plus Hilbert's 90) implies that $Pic(G)$ classifies the central extensions of $G$ by the multiplicative group $G_m$, by some stuff in SGA 7, I believe.</p> <p>The way that I can prove the above fact is by using Kottwitz's isomorphism, which describes $Pic(G)$ in terms of the dual group. I'll probably include this Kottwitzish proof in something I'm writing, but I'm left with the following question:</p> <p>Is there a proof in the literature that $m^\ast(L)$ is isomorphic to $pr_1^\ast(L) \cdot pr_2^\ast(L)$ for line bundles over reductive groups? Someone must have written this up 30 years ago, right? And the implication that $Pic(G)$ classifies central extensions by $G_m$? Is this published somewhere? It certainly shouldn't require passage to the dual group!</p> <p>Of course, if I've messed something up, and the above fact is false, I'd appreciate such information too! </p> http://mathoverflow.net/questions/30265/reference-for-picg-and-central-extensions/30349#30349 Answer by Mikhail Borovoi for Reference for Pic(G) and central extensions. Mikhail Borovoi 2010-07-02T20:29:55Z 2010-07-02T20:29:55Z <p>I can give a reference only for the second part of the question, namely, about central extensions. It was answered by Colliot-Thélène in 2008, not 30 years ago! Colliot-Thélène's paper <em>Résolutions flasques des groupes linéaires connexes</em>, J. für die reine und angewandte Mathematik (Crelle) 618 (2008), 77--133, contains the following corollary (I type it in English):</p> <p><strong>Corollary 5.7.</strong> <em>Let $G$ be a connected linear algebraic group, assumed reductive if char $k > 0$. For any smooth $k$-group of multiplicative type $S$, the natural arrow Ext$(G,S)\to$ ker$[H^1(G,S)\to H^1(k,S)]$ is an isomorphism</em>.</p> <p>Taking $S=\mathbf{G}_m$, we obtain $H^1(k,\mathbf{G}_m)=1$ (Hilbert 90) and $H^1(G,\mathbf{G}_m)=\mathrm{Pic}(G)$ (here $H^1$ means étale cohomology). We obtain an isomorphism Ext$(G,\mathbf{G}_m)\cong \mathrm{Pic}(G)$.</p>