Sorry in advance if my question does not have the required level. Let $K$ be a commutative ring (let say an integral domain for simplicity) and let $0\neq s\in K$. Let $K[s^{-1}]$ be the localized ring. When $K[s^{-1}]$ is a compact object in the derived category $D(K)$ ?
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In short: essentially only when $s$ is idempotent (up to multiplication by a unit), otherwise $K[s^{-1}]$ is not even finitely generated as a module. Obviously, if $K$ is a domain then $s = 0$ or $1$ and in these cases $K[s^{-1}] = K$ or the trivial module $0$.
A complex of $K$-modules is compact if and only if it is perfect, i.e. quasi-isomorphic to a bounded complex of projective modules. For a single module this means that it has a bounded resolution of projective, finitely generated, modules. In particular, the module has to be finitely generated.
answered Jun 20, 2017 at 22:47