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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)?

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  • $\begingroup$ Just if $T'$ is generated by a set of compact objects. $\endgroup$
    – Fernando Muro
    Commented Feb 3, 2021 at 13:28
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    $\begingroup$ Theorem 7.2.1 in Krause, Henning. “Localization Theory for Triangulated Categories.” In Triangulated Categories, edited by Thorsten Holm, Peter Jorgensen, and Raphael Rouquier, 161–235. Cambridge: Cambridge University Press, 2010. doi.org/10.1017/CBO9781139107075.005. $\endgroup$
    – Fernando Muro
    Commented Feb 3, 2021 at 13:31
  • $\begingroup$ @FernandoMuro Thank you. That is exactly my question: what kind of criterion we can use for compact generation of T'. $\endgroup$
    – LocalizeAndConquer
    Commented Feb 3, 2021 at 13:46
  • $\begingroup$ By inspection. In general, that question is far from trivial, it is connected to the telescope conjecture, for instance. $\endgroup$
    – Fernando Muro
    Commented Feb 3, 2021 at 14:40
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    $\begingroup$ It might happen that $T^c \cap T'$ is $0$. $\endgroup$
    – Leo Alonso
    Commented Feb 3, 2021 at 15:06

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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.

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  • $\begingroup$ Thanks! So examples can be taken where T' is the essential image of a functor that does not commute with coproducts. Probably such a thing won't be localizing in general but it is in this case? $\endgroup$
    – LocalizeAndConquer
    Commented Feb 3, 2021 at 15:36
  • $\begingroup$ There are some properties that are probably special to this case (this localizing subcategory is essentially the kernel of "profinite completion"). $\endgroup$
    – Tyler Lawson
    Commented Feb 3, 2021 at 15:57
  • $\begingroup$ (I'm not sure on your comment about coproducts, though; the inclusion $D(\Bbb Q) \to D(\Bbb Z)$ does commute with those?) $\endgroup$
    – Tyler Lawson
    Commented Feb 3, 2021 at 16:02
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    $\begingroup$ Thanks - and Sorry, got my adjointness confused, I meant it doesnt preserve compact objects. $\endgroup$
    – LocalizeAndConquer
    Commented Feb 3, 2021 at 16:03
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    $\begingroup$ Note that here $T'$ is compactly generated, but that the compact objects of $T'$ are not compact in $T$. Note also that in other examples it can happen that $T^c\cap T'$ contains nontrivial objects but they are not compact in $T'$, and it can also happen that the only compact object of $T'$ is zero. There are a number of examples in Appendix B of "Morava K-theories and localisation". $\endgroup$
    – Neil Strickland
    Commented Feb 3, 2021 at 16:27

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