Approximability and Rouquier dimension for noncommuative algebras over schemes

Abstract

This work is concerned with approximability (via Neeman) and Rouquier dimension for triangulated categories associated to noncommutative algebras over schemes. Amongst other things, we establish that the category of perfect complexes of a coherent algebra over a separated Noetherian scheme is strongly generated if, and only if, there exists an affine open cover where the algebra has finite global dimension. As a consequence, we solve an open problem posed by Neeman. Further, as a first application, we study the existence of generators and behaviour under the derived pushforward for Azumaya algebras.