Can $Ch(M)$, for $M$ the category of $R$-modules, be a category of $S$ modules? Can it be the category of quasi-coherent sheaves on some scheme? Let $Ch(M)$ be category of chain complexes of modules in $M$, for $M$ the abelian category of $R$-modules (and $R$ a commutative unital ring).
Can $Ch(M)$ be a category of $S$ modules for some (not graded) commutative unital ring $S$? (My guess is no, because I don't think that there is a generator for the category $Ch(M)$.)
Can it be the category of quasi-coherent sheaves on some scheme?
(Can it be the category of sheaves of $O_X$ modules on some ringed space?)
I'm just curious about how these different classes of abelian categories might differ from each other, and what sorts of computations a person could do to see this difference.
 A: Here's another argument that $Ch(M)$ can't be the category of modules over any ring.  For each $n$, let $M_n$ be any chain complex whose only nonzero term is in degree $n$.  Then in $Ch(M)$, the natural map $\coprod_n M_n\to \prod_n M_n$ from the coproduct to the product is an isomorphism.  This cannot happen for nonzero modules over a ring, so $Ch(M)$ cannot be equivalent to the category of modules over any ring.
A: A precise characterization of module categories $\text{Mod}(R)$ among abelian categories is known (due to Gabriel). We don't need its full strength, but we do need the following necessary condition: there must be an object (namely $R$) which is not only a generator but which is compact projective, or tiny, meaning that $\text{Hom}(R, -)$ preserves all colimits. 
General categorical results (detailed in the second blog post linked to above) can be used to characterize the tiny objects in $\text{Ch}(\text{Mod}(R))$; in particular, they are bounded, and no bounded chain complex can be a generator. (Note that nowhere in this argument do I assume that $R$ is commutative. To characterize the module categories over commutative rings you just need the additional condition that the endomorphism ring of the generator happens to be commutative.) 
