Reference for Pic(G) and central extensions.

2010-07-02T03:47:57Z

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 $G$.

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.

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:

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!

Of course, if I've messed something up, and the above fact is false, I'd appreciate such information too!

Answer by Mikhail Borovoi for Reference for Pic(G) and central extensions.

2010-07-02T20:29:55Z

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 Résolutions flasques des groupes linéaires connexes, J. für die reine und angewandte Mathematik (Crelle) 618 (2008), 77--133, contains the following corollary (I type it in English):

Corollary 5.7. 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.

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$(G,\mathbf{G}_m)\cong \mathrm{Pic}(G)$.

Reference for Pic(G) and central extensions. - MathOverflow

Reference for Pic(G) and central extensions.

Answer by Mikhail Borovoi for Reference for Pic(G) and central extensions.