Approximation by perfect complexes detects Rouquier dimension
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917894154092544 |
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| author | Lank, Pat Olander, Noah |
| author_facet | Lank, Pat Olander, Noah |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10146 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation by perfect complexes detects Rouquier dimension Lank, Pat Olander, Noah Algebraic Geometry Commutative Algebra 14A30 (Primary), 14F08, 13D09, 18G80, 14B05 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. |
| title | Approximation by perfect complexes detects Rouquier dimension |
| topic | Algebraic Geometry Commutative Algebra 14A30 (Primary), 14F08, 13D09, 18G80, 14B05 |
| url | https://arxiv.org/abs/2401.10146 |