When does cohomology of a pro-algebraic group commute with filtered colimits of coefficients?

Question

Let $G$ be a pro-algebraic group, that is, a projective limit of algebraic groups $G_i$. Let $V_i$ be an inductive system of finite dimensional rational $G_i$-representations, so that the colimit $V$ is a rational $G$ representation.

What conditions are required on $G$ to ensure that $$ H^n(G,V)\simeq \mathrm{colim}_iH^n(G, V_i)? $$ I am specifically interested in the case where $G$ is the pro-algebraic completion of a discrete group $\Gamma$. In this case, what conditions are required of $\Gamma$ for the above isomorphism to hold?

Answer 1

This isomorphism always holds, and no conditions are needed. Here I presume that the pro-algebraic group $G$ is pro-affine (as the context of the question seems to suggest).

Let $G$ be a pro-affine pro-algebraic group over a field $k$. Denote by $C=O(G)$ the ring of regular functions on $G$. What is important for us is not the ring but the coalgebra structure on $C$, which is induced by the group structure on $G$. So $C$ is a coassociative, counital, not necessarily cocommutative coalgebra over $k$.

A (rational) representation of $G$ over $k$ is the same thing as a $C$-comodule. (It does not matter whether to consider the left or right $C$-comodules, as the antipode map induced by the inverse element map $g\mapsto g^{-1}\colon G\longrightarrow G$ provides an isomorphism between $C$ and its opposite coalgebra.)

Generally, let $C$ be a coassociative, counital coalgebra over aa field $k$, and let $M$ be a left $C$-comodule. The $k$-vector spaces of cohomology of $C$ with the coefficients in $M$, defined as the $\operatorname{Ext}$ spaces $H^*(C,M)=\operatorname{Ext}_C^*(k,M)$ taken in the category of left $C$-comodules ($=$ discrete/rational modules over the $k$-algebra $C^*$ dual to $C$), are computable as the cohomology spaces of the cobar-complex

$$ M\longrightarrow C\otimes_kM\longrightarrow C\otimes_kC\otimes_kM\longrightarrow C\otimes_kC\otimes_kC\otimes_kM\longrightarrow\dotsb $$

As it is clear from the form of this complex, its cohomology, viewed as a functor of $M$, commutes with the (filtered) direct limits. So for any (directed) inductive system of left $C$-comodules $M_i$, we have $H^*(C,\varinjlim_i M_i)=\varinjlim_iH^*(C,M)$.

Returning to the case of a pro-affine proalgebraic group $G$ and its coalgebra $C=O(G)$, the cohomology of $G$ with the coefficients in a (rational) $G$-module $V$ is the same thing as the cohomology of the coalgebra $C$ with the coefficients in the $C$-comodule $V$, that is $H^*(G,V)=H^*(C,V)$. Thus the functor $H^*(G,{-})$ preserves filtered direct limits.