Let $G$ be a smooth group scheme over some base $S$. Then we have the $S$-stack $BG$ whose $T$-points are the $G$-torsors on $T$. Under which conditions do we have $\mathsf{Qcoh}(BG) \simeq \mathrm{Rep}_S(G)$? I think it holds when $G$ is affine and étale over $S$. I am interested in more general assumptions and references to the literature (because I don't know any). I am sure that this is well-known, but I'm not sure where to look.
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asked May 4, 2014 at 20:13
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$\begingroup$ You should have a look at Laumon and Moret-Bailly, 13.3.6 and 13.3.7. $\endgroup$– Matthieu RomagnyCommented May 4, 2014 at 21:17
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$\begingroup$ Ok they give this as an exercise / without explanation. $\endgroup$– Martin BrandenburgCommented May 5, 2014 at 10:45
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1 Answer
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It would help to specify what topology you are choosing on schemes over $S$ when you define the stack $BG$. Since you specify that $G \to S$ is a smooth surjection, let's assume smooth surjections are covers in our topology. Smooth surjections are stable effective descent morphisms for quasicoherent sheaves, so pullback along the cover $S \to BG$ induces an equivalence of categories between quasicoherent sheaves on $BG$ and $G$-equivariant quasicoherent sheaves on $S$ (equivalently, representations of $G$). That is, there are no additional conditions necessary beyond your initial assumption.
More generally, you can take $G \times_S G \to G \leftrightarrows S$ any fpqc group object, as long as $BG$ is viewed as a stack in the fpqc topology.
answered May 4, 2014 at 23:48
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$\begingroup$ Thanks! Can you give a reference where all this is proven in detail? (It's not that I don't understand it, but rather I need a good reference in my thesis.) $\endgroup$ Commented May 5, 2014 at 10:43
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$\begingroup$ The equivalence in question is mostly a special case of stacks.math.columbia.edu/tag/06WT where $U=S$ and $R=G$. I do not have a good single reference for the fact that you can weaken fppf to fpqc by taking a suitable stackification of $BG$, but it follows from descent properties of quasi-coherent sheaves (stacks.math.columbia.edu/tag/023T ). $\endgroup$– S. Carnahan ♦Commented May 5, 2014 at 18:12
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$\begingroup$ Ok that's the first part, but how to come from G-equiv sheaves to G-reps? Basically I know how this works, but again I don't know a reference to the literature. $\endgroup$ Commented May 5, 2014 at 18:14
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1$\begingroup$ @MartinBrandenburg arxiv.org/abs/math/0412512 Proposition 3.48, setting $X=S$ and $\mathcal{F}$ to be the fibered category of quasicoherent sheaves. $\endgroup$– S. Carnahan ♦Commented May 5, 2014 at 18:39
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$\begingroup$ Thank you. Honestly, I'm very confused. But I accept your answer because probably others will benefit from the references. $\endgroup$ Commented May 5, 2014 at 19:34