Approximation by perfect complexes detects Rouquier dimension
Pat Lank, Noah Olander
TL;DR
The paper addresses bounding the Rouquier dimension of $D^b_{ ext{coh}}(X)$ for Noetherian schemes by linking the ability to build all perfect complexes from a generator $G$ within $d$ cones to already constructing the entire derived category within $d$. It develops a framework of approximations by perfect complexes in triangulated categories with bounded $t$-structure, proving that if $\operatorname{perf}X \subseteq \langle G \rangle_d$ and $D^b_{ ext{coh}}(X) = \langle G \rangle_d$, then strong generation follows and practical dimension bounds arise. The work yields sharp results such as invariance of Rouquier dimension under étale extensions for affine schemes with a dualizing complex, a bound of $\dim D^b_{ ext{coh}}(X) \le 2$ for curves whose delta invariant at closed points is at most one, and a bound for birational derived splinter varieties via a resolution of singularities. Collectively, these findings provide a versatile, approximation-based toolkit for analyzing Rouquier dimension in singular and non-projective settings, with implications for birational geometry and the study of derived categories of schemes.
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 étale extensions and is invariant under étale 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.