  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.
