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Is there a nice analog of the Freyd-Mitchell theorem for triangulated categories (potentially with some requirements)? Freyd-Mitchell is the theorem which says that any small abelian category is a fully faithful, exact embedding into the module category of some ring.

Therefore, I'd like a theorem like this: Any small triangulated category is a fully faithful, triangulated subcategory of the unbounded derived category of modules on some ring.

My guess is that this fails to be true, for similar reasons to a triangulated category not always being the derived category of its core. Is there a simple example of this? -and- Is there a set of properties that do imply the above theorem?

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3 Answers

up vote 8 down vote accepted

There are some things like what you ask for but as Tyler points out one needs restrictions on the categories one can consider.

Any algebraic triangulated category which is well generated is equivalent to a localization of the derived category of a small DG-category - this is a theorem of Porta (ref is M. Porta, The Popescu-Gabriel theorem for triangulated categories. arXiv:0706.4458). Here algebraic (in the sense of Keller) means that the category is equivalent as a triangulated category to the stable category of a Frobenius category (Schwede has a paper on this as well giving conditions in terms of Koszul type objects).

An answer (maybe closer to what you ask) is the following. If one has a Grothendieck abelian category then Gabriel-Popescu tells you it comes from a torsion theory on some category of modules. It turns out this lifts to the level of derived categories so one can view the derived category of a Grothendieck abelian category as a localization of the derived category of R-modules for some R (in particular it comes with a fully faithful embedding into the derived category of R-modules).

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The stable homotopy category isn't equivalent to the derived category of a ring, and there are numerous other "homotopical" triangulated categories that definitely don't come from algebra. There are also some strange algebraically-derived examples that aren't derived categories.

See Muro-Schwede-Strickland's "Triangulated categories without models" (arXiv) for some nice discussion.

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Don't know if this is germane, but Freyd actually proved that every small triangulated category has an abelian envelope. That is, you can embed a triangulated category into an abelian category, in fact a Frobenius category, where the triangulated category is the category of projectives=injectives. I believe he introduced this construction in the same paper, in 1965 or so, in some La Jolla conference proceedings, in which he introduced the generating hypothesis in stable homotopy theory. It also appears in Amnon Neeman's book on triangulated categories, I am pretty sure.

Mark

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