Triangulated categories with a single compact generator, and two Brown representability theorems

We generalize theorems of Bondal and Van den Bergh and of Rouquier. A corollary of our main results says the following. Let X be a scheme proper over a an excellent, finite-dimensional noetherian ring R. Then the Yoneda pairing, taking an object A in the category Dperf(X) and an object B in the category Dcohb(X), to the finite R–module Hom(A,B), gives an equivalence of Dcohb(X) with the category of finite R–linear homological functors H:Dperf(X)op⟶R–mod, and an equivalence of Dperf(X)op with the category of finite homological functors H:Dcohb(X)⟶R–mod. Recall: a homological functor H is finite if ⊕Hii=−∞∞(C) is a finite R–module for every C∈Dperf(X). Bondal and Van den Bergh proved the special case, of the assertion about Dcohb(X) identifying with the finite homological functors on Dperf(X)op, as long as R is a field and X is projective over R. And the assertion about Dperf(X)op, identifying with the finite homological functors on Dcohb(X), again under the assumption that X is projective over a field R, is due to Rouquier. But our theorems are far more general yet. They aren’t only about schemes, they work in the abstract generality of triangulated categories with coproducts and a single compact generator, satisfying a certain approximability property. At the moment I only know how to prove this approximability for the categories Dqc(X) with X a quasicompact, separated scheme, for the homotopy category of spectra, for the category D(R) where R is a (possibly noncommutative) negatively graded dg algebra, and for certain recollements of the above. The work was inspired by Jack Hall’s elegant new proof of a vast generalization of GAGA, a proof based on representability theorems of the type above. The generality of Hall’s result made me wonder how far the known representability theorems could be improved.