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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!

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Marty, Raynaud's thesis was closer to 40 years ago. :) There is a very elegant proof of the relation with central extensions by using the "big cell" structure over $F_s$ (no need to assume $F$ perfect) and Rosenlicht's result relating units on algebraic groups to homomorphisms to $\mathbf{G}_ m$. No need to refer to SGA7 (Rosenlicht's unit result is discussed in there, albeit without proof). The crux is that in the simply connected case the coordinate ring is a UFD, a fact that seems not as well-known as it should be. It is easier to discuss this in person; are you around UCSC these days? –  BCnrd Jul 2 '10 at 5:48
    
Great reference - Thanks! I haven't quite found the result I was looking for, word-for-word, in Raynauds thesis (LNM 119, I presume). But I suppose it follows from something in there, by viewing $G$ as the homogeneous space $G \times G / \Delta(G)$. I'll peruse some more and try to find exactly what I'm looking for. I'm in Australia right now (!), back in CA on July 11. Perhaps we can talk in person in mid-late July, if you'll be around. –  Marty Jul 2 '10 at 7:27
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Marty, the 2nd half of p. 110 and Remark on the top of p. 111 of LNM 119 were what I had in mind, but I now see they don't quite address your question. A better reference, over alg. closed fields, is section 4 of "Local properties of algebraic group actions" by Knop, Kraft, Luna, & Vust. In late July we can discuss why their idea works (for connected ss gps) over a general field if you don't see it. (Can ignore most of that section 4 for your purposes.) I will now email you .pdf's of this and of a scan of an awe-inspiring letter I got from Gabber on a (surprise!) vast generalization. –  BCnrd Jul 2 '10 at 12:18

1 Answer 1

up vote 2 down vote accepted

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

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Thank you - I was unaware of this paper of Colliot-Thelene, and it seems to answer both my questions on the nose. Corollary 5.7, as you mention, answers the second part of my question. And the first part of my question is answered earlier, right after Thm 5.6 in the Colliot-Thelene paper. In light of this, and BCnrd's comments and references, I'll give my own Kottwitz-inspired argument, reference to the C-T paper, and something like "this seems to have been known to the experts (Raynaud, Gabber) for almost 40 years". –  Marty Jul 3 '10 at 1:43
    
One more thank you... I've found your papers on "Abelian Galois Cohomology..." and "Extended Picard Complexes..." very helpful throughout. –  Marty Jul 3 '10 at 1:45
    
Ofer was certainly an early starter, but I'm not so sure he would have known too much about Picard groups 40 years ago... –  Kevin Buzzard Jul 3 '10 at 21:20

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