Categories which are not compactly generated - MathOverflow

Categories which are not compactly generated

2009-11-11T01:11:40Z

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)

Answer by Mariano Suárez-Alvarez for Categories which are not compactly generated

2009-11-11T01:38:29Z

For some rings $R$, Brown representability fails in the derived category $\mathcal{D}(R)$, so the latter is not compactly generated. There is a nice paper by Christiansen, Keller and Neeman on the subject, where they construct examples.

Answer by Reid Barton for Categories which are not compactly generated

2009-11-11T01:44:30Z

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.

Answer by Matthew Ballard for Categories which are not compactly generated

2009-11-11T02:54:07Z

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.

Answer by Greg Stevenson for Categories which are not compactly generated

2009-11-11T02:56:52Z

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.

Answer by Mark Hovey for Categories which are not compactly generated

2009-11-11T15:49:18Z

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