The category $\big[{\mathcal T}^c\big]^{\text{op}}$ as functors on ${\mathcal T}^b_c$

Abstract

We revisit an old assertion due to Rouquier, characterizing the perfect complexes as bounded homological functors on the bounded complexes of coherent sheaves. The new results vastly generalize the old statement---first of all the ground ring is not restricted to be a field, any commutative, noetherian ring will do. But the generalization goes further, to the abstract world of approximable triangulated categories.