Semi-orthogonal decompositions of GIT quotient stacks

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