Let $k$ be a ring (resp. profinite ring), $G$ a group (resp. profinite group), and $k[G]$ the group algebra (resp. completed group algebra).


For any such $G$, we may associate to it the group of units $k[G]^\times$ of $k[G]$, and this association is clearly functorial. Has this functor been studied at all?


For example, given an exact sequence
$$1\rightarrow G\rightarrow G'\rightarrow G''\rightarrow 1$$
we get a sequence
$$1\rightarrow k[G]^\times/k^\times\rightarrow k[G']^\times/k^\times\rightarrow k[G'']^\times/k^\times\rightarrow 1$$
which is probably not exact, but at least the map $k[G]^\times/k^\times\rightarrow k[G']^\times/k^\times$ is injective, and so one might hope that the functor $G\mapsto k[G]^\times$ is left exact. 


If it's left-exact, is this actually exact? If not, is anything known about its right-derived functors?


I suppose this is best posed under the assumption that $G$ is abelian, though I'm also very interested in the nonabelian case (or at least whatever still makes sense there in terms of cohomology)



What is known about the functor $G\mapsto k[G]^\times/k^\times$?
Forget about exactness; this functor isn't *additive*, which pretty much tanks any hope of doing homological algebra to it even if you restrict to abelian groups. 


Here's a special case that's easy to understand. Suppose $G$ is finite and $k$ has characteristic not dividing $|G|$. Then $k[G]$ is semisimple, so we have a decomposition


$$k[G] \cong \prod\_i M\_{n\_i}(D\_i)$$


where the product is indexed over the irreducible representations $V\_i$ of $G$ over $k$, with $\text{End}\_G(V\_i) = D\_i$ division algebras and $n\_i = \dim\_{D\_i} V\_i$. Hence


$$k[G]^{\times} \cong \prod\_i GL\_{n\_i}(D\_i)$$


and quotienting by $k^{\times}$ just removes the trivial representation. The functoriality with respect to morphisms $f : G \to H$ comes from pulling back irreps of $H$ to $G$ and seeing how they decompose. 


$k[G]^{\times}$ naturally occurs as the automorphism group of the forgetful functor from $G$-representations to $k$-vector spaces (regarded just as a functor, not as a monoidal functor). 


