TITLE: Is K(R-Mod) compactly generated when R is an artin algebra?
QUESTION [5 upvotes]: I wonder if the triangulated category K(R-Mod) is compactly generated when R is an artin algebra? R-Mod denotes all left R-modules. I understand this would be true if R has finite representation type since R-modules then are direct sums of finitely generated ones, but I am interested in the general case. Could it be that a generating set are finitely generated R-modules and shifts of them. (This would not be true for general rings, e.g. Neeman showed that K(Z-mod) is not compactly generated.)
Thanks.

REPLY [2 votes]: The answer is in general no - $K(R\text{-}\mathrm{Mod})$ can fail to be well generated even when $R$ is artinian. As you mention $K(R\text{-}\mathrm{Mod})$ is compactly generated if $R$ is of finite representation type. It turns out that the converse holds. This is a result of Jan Šťovíček which occurs as Proposition 2.6 in  this paper. The precise result is:
Proposition Let $R$ be a ring. The following are equivalent:
(i) $K(R\text{-}\mathrm{Mod})$ is well generated;
(ii) $K(R\text{-}\mathrm{Mod})$ is compactly generated;
(iii) $R$ is left pure semisimple.
In particular, when $R$ is artinian this occurs precisely when $R$ has finite representation type.