TITLE: What is known about the functor $G\mapsto k[G]^\times/k^\times$?
QUESTION [6 upvotes]: 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)

REPLY [6 votes]: 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).