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 ${\mathbf {D}}^{\mathrm {perf}}(X)$ and an object $B$ in the category $\mathbf {D}^{b}_{\mathrm {coh}}(X)$ , to the finite $R$ –module $\mathop {\mathrm {Hom}}(A,B)$ , gives an equivalence of $\mathbf {D}^{b}_{\mathrm {coh}}(X)$ with the category of finite $R$ –linear homological functors $H:{\mathbf {D}}^{\mathrm {perf}}(X)^{\mathrm {op}}\longrightarrow \mathop {R\textup {--mod}} \ $ , and an equivalence of ${\mathbf {D}}^{\mathrm {perf}}(X)^{\mathrm {op}}$ with the category of finite homological functors $H:\mathbf {D}^{b}_{\mathrm {coh}}(X)\longrightarrow \mathop {R\textup {--mod}} \ $ . Recall: a homological functor $H$ is finite if $\oplus _{i=-\infty }^{\infty }H^{i}(C)$ is a finite $R$ –module for every $C\in {\mathbf {D}}^{\mathrm {perf}}(X)$ . Bondal and Van den Bergh proved the special case, of the assertion about $\mathbf {D}^{b}_{\mathrm {coh}}(X)$ identifying with the finite homological functors on ${\mathbf {D}}^{\mathrm {perf}}(X)^{\mathrm {op}}$ , as long as $R$ is a field and $X$ is projective over $R$ . And the assertion about ${\mathbf {D}}^{\mathrm {perf}}(X)^{\mathrm {op}}$ , identifying with the finite homological functors on $\mathbf {D}^{b}_{\mathrm {coh}}(X)$ , 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 ${\mathbf{D}_{\text{\textbf{qc}}}}(X)$ with $X$ a quasicompact, separated scheme, for the homotopy category of spectra, for the category ${\mathbf{D}}(R)$ 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.
