3
votes
1answer
203 views

Quasi-coherent sheaves on classifying stacks

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 ...
1
vote
1answer
194 views

Smooth map to the stack of G-bundles

Let $G$ a semisimple group and $B$ a Borel subgroup. We denote by $Bun_{G}$ the stack of G-bundles. Is it true that a certain open subset $Bun_{B,r}$ maps smoothly to $Bun_{G}$? My question comes ...
8
votes
1answer
450 views

Recovering classical Tannaka duality from Lurie's version for geometric stacks

In Lurie's paper Tannaka Duality for Geometric Stacks, it is essentially shown that specifying a morphism of geometric objects $$ f \colon X \to Y$$ is equivalent to giving a corresponding pullback ...
11
votes
4answers
711 views

Does every morphism BG-->BH come from a homomorphism G-->H?

Given a homomorphism f:G→H between smooth algebraic groups, we get an induced homomorphism of algebraic stacks Bf:BG→BH, given by sending a G-torsor P over a scheme X to the H-torsor PxGH, ...
18
votes
4answers
1k views

algebraic group G vs. algebraic stack BG

I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...