  Let $Y$ be a singular algebraic variety and let $\TY$ be a resolution of
singularities of $Y$. Assume that the exceptional locus of $\TY$ over $Y$ is an
irreducible divisor $\TZ$ in $\TY$. For every Lefschetz decomposition of $\TZ$
we construct a triangulated subcategory $\TD \subset \D^b(\TY)$ which gives a
desingularization of $\D^b(Y)$. If the Lefschetz decomposition is generated by
a vector bundle tilting over $Y$ then $\TD$ is a noncommutative resolution, and
if the Lefschetz decomposition is rectangular, then $\TD$ is a crepant
resolution.
