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?



Freyd-Mitchell for triangulated categories?
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).


