Moscow Mathematical Journal

Volume 25, Issue 1, January–March 2025 pp. 13–31.

Approximation by Perfect Complexes Detects Rouquier Dimension

Authors: Pat Lank (1) and Noah Olander (2)
Author institution:(1) Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.
(2) Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG, Amsterdam, Netherlands

Summary:

In this paper we study 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.

2020 Math. Subj. Class. 14A30 (Primary), 14F08, 13D09, 18G80, 14B05.

Keywords: Derived categories, bounded t-structures, approximation by perfect complexes, Rouquier dimension, strong generators, coherent sheaves, derived splinters, étale morphisms.

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