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Do you know natural examples of triangulated categories (or [presentable] stable $\infty$-categories) which are not compactly generated? (ideally they'd be defined algebraically, but curious to hear any examples.. the ones I know are gotten as opposites of compactly generated categories or by slightly ad hoc geometric constructions)

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As David says, D(R) is compactly generated. This means Brown representability for COHOMOLOGY is automatically true, but that does NOT mean Brown representability for HOMOLOGY is true, and in fact it is not always true. That is what the Christensen-Keller-Neeman example shows.

Let K(R) be the category of chain complexes over R and chain homotopy classes of chain maps. Then K(Z) is not compactly generated (and K(R) is not either for many R). This is proved in a paper of mine with Christensen.

Quillen model structures for relative homological algebra

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  • $\begingroup$ K(R) is a very interesting example, thanks! but am I correct to assume it's not stable? $\endgroup$
    – David Ben-Zvi
    Commented Nov 12, 2009 at 18:31
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    $\begingroup$ No, K(R) is stable. I was using unbounded chain complexes, and the suspension is the usual shift, which is obviously an equivalence. The reason we don't use K(R), but instead use D(R), is precisely that K(R) is not compactly generated--there are nontrivial objects of K(R) with no homology. $\endgroup$
    – Mark Hovey
    Commented Nov 13, 2009 at 13:20
  • $\begingroup$ Sorry for the dumb questions, but what is K(R)? What is D(R)? Also, you guys keep talking about Brown representability; how is that related to compact generators? $\endgroup$
    – Kevin H. Lin
    Commented Nov 16, 2009 at 13:54
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    $\begingroup$ Ch(R) is the category of unbounded chain complexes of R-modules, for an ordinary ring R, where d goes down a degree. K(R) is the category obtained from Ch(R) by identifying chain homotopic maps. D(R) (the derived category) is the category obtained from Ch(R) (or K(R)) by formally inverting homology isomorphisms. Both K(R) and D(R) are triangulated, though Ch(R) is abelian. I mentioned Brown representability because of the answer of Mariano. If you knew Brown representability for cohomology failed, then your category could not be compactly generated. $\endgroup$
    – Mark Hovey
    Commented Nov 17, 2009 at 18:18
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Not from modular representation theory, but the following seems like a geometrically natural example:

Neeman has a nice note where he shows that the only compact object, in the derived category of sheaves of abelian groups on a non-compact, connected, positive-dimensional manifold, is the zero object.

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    $\begingroup$ Good example ;) $\endgroup$
    – Greg Stevenson
    Commented Nov 11, 2009 at 2:58
  • $\begingroup$ Presumably the same argument works for sheaves of Q-vector spaces, which answers the question in my answer. $\endgroup$
    – Reid Barton
    Commented Nov 11, 2009 at 3:08
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One example is the following - suppose that $M$ is a non-compact connected manifold of dimension $\geq 1$. Then the unbounded derived category of chain complexes of sheaves of abelian groups on $M$ has no compact objects other than zero. It is, however, well generated. This can be found in Neeman's paper "On the Derived Category of Sheaves on a Manifold".

Another example is suppose that $R$ is the ring $k\oplus V$ where $k$ is a field, $V$ is an infinite dimensional vector space and the ring structure is the unique one making $V$ a square zero ideal. Then the category $K(Proj\; R)$, the homotopy category of complexes of projective $R$-modules is not compactly generated. Again though it is well generated (even $\aleph_1$-compactly generated). In general this category need not be compactly generated if $R^{op}$ is not coherent.

One could also produce more examples along these lines I imagine by considering for instance the homotopy category of flat modules over a suitable ring.

These last two examples are very natural objects to study - I can provide more details on why if anyone is interested.

These are the most naturally occurring examples I know of off the top of my head.

I'll try and think of a modular representation example if I can although when I think about these things it is normally the stable category for modular reps of a finite group and these are always compactly generated.

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There is also, now a bunch of years later, a result of Hall, Neeman, Rydh that says over a field of char $p$,there are no non trivial compact objects in $D_{qc}(B\mathbb{G}_a)$.

Moreover they prove for any algebraic stack ,$X$, which is quasi-compact, quasi-separated and "poorly stabilized" (roughly that there exists a point whose stabilizer contains a copy of $\mathbb{G}_a$) then $D_{qc}(X)$ is never compactly generated.

https://arxiv.org/pdf/1405.1888.pdf

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Funny, we were just discussing this at dinner last night. Anyways, see Corollary B.13 of Morava K-theories and localization by Hovey and Strickland for some stable presentable ∞-categories with no nonzero compact (they call them small) objects, hence not compactly generated. One example is the H$\mathbb{F}_p$-local category. Also see my question here for related discussion.

I would like to know whether there are any examples which are H$\mathbb{Q}$-enriched (modules over H$\mathbb{Q}$-mod). I think there should be, but I don't know exactly how to write one down.

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  • $\begingroup$ Thanks for that reference! seems totally counterintuitive to me, since I'm used to the "classical" world where looking at say p-local modules would be modules over a ring (localization at p) hence compactly generated.. do you have any intuition for these results you could share? $\endgroup$
    – David Ben-Zvi
    Commented Nov 11, 2009 at 2:15
  • $\begingroup$ For HQ-enriched results, are you looking for ones with no compact objects? there I have no idea.. for things that just are not compactly generated, you can look at eg opposite categories of your favorite compactly generated categories (eg pro-perfect modules over a Q-algebra). A more geometric example comes I think (from MY dinner conversation) from taking an infinite chain of affine lines glued pairwise.. $\endgroup$
    – David Ben-Zvi
    Commented Nov 11, 2009 at 2:16
  • $\begingroup$ I'll ask a more specific followup - seeing if there are examples from modular rep theory.. $\endgroup$
    – David Ben-Zvi
    Commented Nov 11, 2009 at 2:17
  • $\begingroup$ Just not compactly generated would be sufficient. The opposite of a presentable category is basically never presentable, so those examples don't qualify. And I don't have any real intuition for the results I cited, except to note that HF_p /\ HF_p is very large (the dual Steenrod algebra) in contrast to F_p ⊗ F_p, so it's not too surprising that HF_p-local things are not the same as HF_p-modules. $\endgroup$
    – Reid Barton
    Commented Nov 11, 2009 at 2:57
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    $\begingroup$ Even if you're looking at the derived category of Z-modules, F_p-local things (in the sense that it's used in Hovey-Strickland) are not F_p-modules, but p-complete modules. Localization in this sense with respect to a quotient (rather than subobject) behave like a formal completion. $\endgroup$
    – Tyler Lawson
    Commented Nov 11, 2009 at 4:05

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