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arXiv · 2011.12743

Stably semiorthogonally indecomposable varieties

Dmitrii Pirozhkov
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Abstract

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) scheme. This property implies, among other things, that each connected closed subscheme whose structure sheaf is a perfect complex 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 scheme which admits a finite morphism to an abelian variety is NSSI and that the total space of a fibration over a regular 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.

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Dmitrii Pirozhkov. 2020-11-25. Stably semiorthogonally indecomposable varieties. https://doi.org/10.46298/epiga.2023.volume7.7700

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