Let T be a compactly generated triangulated category and let T' be a localizing subcategory. Is it automatic that T' is comapctly generated by $T^c \cap T'$, where $T^c$ is compact objects of $T$?


Edit: I would be interested if there is a useful sufficient criteria (that takes advantage of the compact generation of T)?



Subcategory of compactly generated triangulated category
Let $T' \subset D(\Bbb Z)$ be the collection of complexes whose homology is uniquely divisible; i.e. $T'$ is the essential image of $D(\Bbb Q)$. Then $T'$ is a localizing subcategory. However, compact objects of $D(\Bbb Z)$ have finitely generated homology groups, and so the only compact objects in $T'$ are zero objects.


