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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2011.12743 |
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| _version_ | 1866910069391622144 |
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| author | Pirozhkov, Dmitrii |
| author_facet | Pirozhkov, Dmitrii |
| contents | A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. We introduce a definition of a noncommutatively stably semiorthogonally indecomposable (NSSI) variety. This propery implies, among other things, that each smooth proper subvariety has indecomposable derived category of coherent sheaves, and that if $Y$ is NSSI, then for any variety $X$ all semiorthogonal decompositions of $X \times Y$ are induced from decompositions of $X$. We prove that any variety whose Albanese morphism is finite is NSSI, and that the total space of a fibration over NSSI base with NSSI fibers is also NSSI. We apply this indecomposability to deduce that there are no phantom subcategories in some varieties, including surfaces $C \times \mathbb{P}^1$, where $C$ is any smooth proper curve of positive genus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_12743 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Stably semiorthogonally indecomposable varieties Pirozhkov, Dmitrii Algebraic Geometry A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. We introduce a definition of a noncommutatively stably semiorthogonally indecomposable (NSSI) variety. This propery implies, among other things, that each smooth proper subvariety has indecomposable derived category of coherent sheaves, and that if $Y$ is NSSI, then for any variety $X$ all semiorthogonal decompositions of $X \times Y$ are induced from decompositions of $X$. We prove that any variety whose Albanese morphism is finite is NSSI, and that the total space of a fibration over NSSI base with NSSI fibers is also NSSI. We apply this indecomposability to deduce that there are no phantom subcategories in some varieties, including surfaces $C \times \mathbb{P}^1$, where $C$ is any smooth proper curve of positive genus. |
| title | Stably semiorthogonally indecomposable varieties |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2011.12743 |