Derived global functions on (derived) stacks $BG$ and $G/G$

Question

In Toen's Affine Stacks, he computes that $\mathcal{O}(B\mathbb{G}_a) = k[\epsilon]$ with $|\epsilon| = 1$ and trivial differential (where here $\mathcal{O}$ is computed in a derived sense, and we choose dg algebras as a model for derived rings). To prove this, he explicitly constructs a cosimplicial ring whose $\text{Spec}$ is $B\mathbb{G}_a$, verifying this by showing they satisfy the same universal property.

I'm wondering if there's a more explicit way to do this calculation. In particular, I would like to compute $\mathcal{O}(BG)$ for various linear algebraic $G$ (unipotent, solvable, semisimple...). I would also like to compute $\mathcal{O}(G/G)$ (the adjoint action).

Here's an idea for an approach, which might be totally wrong. These stacks are Artin stacks, so in particular one can write $BG$ as a colimit of a simplicial diagram in schemes. Then by definition, $\mathcal{O}(BG)$ is a limit of cosimplicial diagram in algebras. Here I'm not exactly sure what I'm doing, but I want to say I can apply some kind of (dual) monoidal Dold-Kan to realize this limit in dg algebras. I think (not sure) one only gets a commutative algebra structure up to homotopy here, but at least taking cohomology, this ends up just being the Hochschild complex for computing the ("rational") cohomology of the (rational) trivial representation of $G$.

Using this, I think one computes that $\mathcal{O}(B\mathbb{G}_m) = k$ (but haven't verified the details), and Jantzen's book (Representations of Algebraic Groups) verifies Toen's result for $\mathbb{G}_a$. Very quickly though, this computation becomes difficult. Are there general results/computations known for unipotent, solvable, or reductive $G$, and can one compute $\mathcal{O}(G/G)$ in a similar way (e.g. using the rational cohomology of $k[G]$ under the adjoint action)? Do you have a preferred way to think about this?

Answer 1

For a smooth groupoid $G_1\rightrightarrows G_0$ , here is a way to compute derived global sections $\mathcal O(X)$ of $\mathcal O_X$, where $X=[G_0/G_1]$.

Observe that $X$ is the homotopy colimit of the nerve $G_\bullet$ of the groupoid. As you said then $\mathcal O(X)$ is the homotopy limit of $\mathcal O(G_\bullet)$. This homotopy limit can indeed be computed via cosimplicial Dold-Kan.

This is particular tells you that, as Pavel wrote in his comment, $\mathcal O(BG)=C^*(G,k)$.

Considering the action groupoid $G\times G\rightrightarrows G$ for the conjugation action you get (as suggested in your question) that $\mathcal O(G/G)=C^*(G,\mathcal O(G))$, where the action of $G$ on $\mathcal O(G)$ comes from the conjugation on $G$.

For a reductive $G$, we know that $C^*(G,V)=V^G$ for any $G$-module $V$. Hence $\mathcal O(BG)=k$ and $\mathcal O(G/G)=\mathcal O(G)^G$ is the algebra of class functions.

As Pavel said in his comment, $\mathcal O(B\mathbb{G}_a)=C^*(\mathbb{G}_a,k)=C^*(Lie(\mathbb{G}_a),k)$. Then $Lie(\mathbb{G}_a)=Free(k)$ is the free Lie algebra generated by $k$. Now for any perfect complex $V$, $C^*(Free(V),k)$ is quasi-isomorphic to the square-zero extension $k\oplus V^*[-1]$. One thus gets back that $\mathcal O(B\mathbb{G}_a)=k[\epsilon]$.