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Let $G$ be an affine group scheme over a field $k$ of characteristic zero.

I understand that when $G$ is algebraic there is an equivalence between the category $\text{Rep}(G)$ of (rational) representations of $G$ and the category $\text{QCoh}(BG)$ of quasicoherent sheaves on the fppq classifying stack $BG$.

I have three questions:

  1. Which subcategory, if any, of $\text{QCoh}(BG)$ corresponds to the category $\text{Rep}_{f}(G)$ of finite dimensional representations of $G$? Does it makes sense to talk about coherent sheaves on $BG$?
  2. Does an equivalence of this form hold when $G$ is not assumed to be algebraic?
  3. How about at the level of bounded derived categories?
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  • $\begingroup$ 2) Such an equivalence certainly still holds, with the same proof, namely by using the fppf cover $\text{Spec} k \rightarrow BG$. $\endgroup$ – Jef Jan 26 at 15:37
  • $\begingroup$ Can you elaborate? $\endgroup$ – Patrick Elliott Jan 27 at 12:55
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In a category $\mathcal{A}$ (with all filtered colimits, like $\text{Rep}(G)$), one can define the notion of a compact object $A \in \mathcal{A}$ as an object for which the functor $\text{Hom}_{\mathcal{A}}(A, -)$ commutes with all filtered colimits. Since objects of the abelian category $\text{Rep}(G)$ are direct sums of their irreducible components, it's easy to verify directly that an object is compact if and only if it is finite dimensional. Therefore, the finite dimensional representations are precisely the compact objects of $\text{Rep}(G)$.

Similar assertions to the above hold at the derived category level. For a regular scheme $X$, the compact objects of the derived category $\text{QCoh}(X)$ are precisely the coherent complexes--that is, bounded complexes which have only coherent cohomology. Note we need to go to the non-bounded level to speak about compact objects, since our category must have all filtered colimits.

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I will answer questions (1) and (3). For (1), finite dimensional $k$-linear representations of $G$ correspond exactly to coherent sheaves on $BG$. A coherent sheaf on $BG$ is described exactly using descent theory + the condition of being Cartesian (Definition 3.7 (ii)).

Now for (3), let us first note that there is a natural functor $$\Psi : D(\operatorname{Rep}(G)) = D(\operatorname{QCoh}(BG)) \to D_{\operatorname{qc}}(BG),$$ where the thing on the right is the derived category of all modules on $BG$ with quasi-coherent cohomology. Furthermore, $\Psi$ restricts to a functor $\Psi^+$ on the level of bounded below objects. In Appendix C of Hall-Neeman-Rydh, it is shown that $\Psi^+$ is an equivalence of categories (in fact for much more general algebraic stacks). In fact their proof first shows that $\Psi^+$ preserves bounded objects and in fact is an equivalence on that level, before taking homotopy limits to pass to bounded below objects. This gives an affirmative answer to (3).

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