  If G is a reductive group which acts on a linearized smooth scheme $X$ then
we show that under suitable standard conditions the derived category of
coherent sheaves of the corresponding GIT quotient stack $X^{ss}/G$ has a
semi-orthogonal decomposition consisting of derived categories of coherent
sheaves of rings on the categorical quotient $X^{ss}/\!/G$ which are locally of
finite global dimension. One of the components of the decomposition is a
certain non-commutative resolution of $X^{ss}/\!/G$ constructed earlier by the
authors.
  As a concrete example we obtain in the case of odd Pfaffians a
semi-orthogonal decomposition of the corresponding quotient stack in which all
the parts are certain specific non-commutative crepant resolutions of Pfaffians
of lower or equal rank which had also been constructed earlier by the authors.
In particular this semi-orthogonal decomposition cannot be refined further
since its parts are Calabi-Yau.
  The results in this paper also complement a result by Halpern-Leistner (and
similar results by Ballard-Favero-Katzarkov and Donovan-Segal) that asserts the
existence of a semi-orthogonal decomposition of the derived category of $X/G$
in which one of the components is the derived category of $X^{ss}/G$.
