Semiorthogonal indecomposability of minimal irregular surfaces

We prove a relative version of the fact that semiorthogonal decompositions of the bounded derived category of coherent sheaves are strongly constrained by the base locus of the canonical linear system. As an application we prove that the derived category of minimal surfaces $X$ with $H^1(X,{\mathcal{O}}_X)\mathrm{\ne }0$ are semiorthogonally indecomposable.