Semi-orthogonal decompositions of GIT quotient stacks

Abstract

If G is a reductive group acting on a linearized smooth scheme X then we show that under suitable standard conditions the derived category D(X-ss/G) of the corresponding GIT quotient stack Xss/G has a semi-orthogonal decomposition consisting of derived categories of coherent sheaves of rings on 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 complement results by Halpern-Leistner, Ballard-Favero-Katzarkov and DonovanSegal that assert the existence of a semi-orthogonal decomposition of D(X/G) in which one of the parts is D(X-ss/G).