Semi-orthogonal decompositions of GIT quotient stacks

2016/03/09 by Špenko, Špela, Bergh, Michel Van den · 1 citation
#13A50 #14L24 #16E35 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1603.02858

Abstract

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.

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