Approximation by perfect complexes detects Rouquier dimension

TL;DR

This paper investigates bounds on the Rouquier dimension in derived categories of coherent sheaves, showing it can be characterized by the number of cones needed for perfect complexes and establishing sharper bounds for singular schemes.

Contribution

It introduces a new characterization of Rouquier dimension via approximations and derives sharper bounds for singular schemes and invariance properties under étale extensions.

Findings

Rouquier dimension is invariant under étale covers of affine schemes with a dualizing complex.

For certain curves, Rouquier dimension is at most two.

Bounds on Rouquier dimension for derived splinter varieties relate to their resolutions.

Abstract

This work explores bounds on the Rouquier dimension in the bounded derived category of coherent sheaves on Noetherian schemes. By utilizing approximations, we exhibit that Rouquier dimension is inherently characterized by the number of cones required to build all perfect complexes. We use this to prove sharper bounds on Rouquier dimension of singular schemes. Firstly, we show Rouquier dimension doesn't go up along \'{e}tale extensions and is invariant under \'{e}tale covers of affine schemes admitting a dualizing complex. Secondly, we demonstrate that the Rouquier dimension of the bounded derived category for a curve, with a delta invariant of at most one at closed points, is no larger than two. Thirdly, we bound the Rouquier dimension for the bounded derived category of a (birational) derived splinter variety by that of a resolution of singularities.