Approximability and Rouquier dimension for noncommutative algebras over schemes

TL;DR

This paper investigates the properties of triangulated categories linked to noncommutative algebras over schemes, establishing conditions for strong generation and solving an open problem in the field.

Contribution

It proves that perfect complexes are strongly generated iff the algebra has finite global dimension on an affine cover, addressing a question by Neeman.

Findings

Strong generation of perfect complexes is equivalent to finite global dimension on an affine cover.

Solved an open problem posed by Neeman regarding triangulated categories.

Studied generators for Azumaya algebras.

Abstract

This work is concerned with approximability (\`{a} la 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 Noetherian quasi-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 for Azumaya algebras.