Stably semiorthogonally indecomposable varieties

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.

17 pages; v3 promised too much about non-smooth or non-proper schemes, see Erratum at the end