Approximation by compact complexes holds for quasi-compact quasi-separated algebraic stacks with quasi-finite diagonal, and this property descends along quasi-finite flat covers.
Approximation by perfect complexes detects Rouquier dimension
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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.
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2025 1verdicts
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Compact approximation and descent for algebraic stacks
Approximation by compact complexes holds for quasi-compact quasi-separated algebraic stacks with quasi-finite diagonal, and this property descends along quasi-finite flat covers.