Semi-orthogonal decompositions of GIT quotient stacks

TL;DR

This paper establishes semi-orthogonal decompositions of derived categories of GIT quotient stacks, involving non-commutative resolutions and specific examples like Pfaffians, extending previous work on categorical structures in geometric invariant theory.

Contribution

It constructs semi-orthogonal decompositions for GIT quotient stacks with components being non-commutative resolutions, including explicit examples like Pfaffians, and relates to existing categorical decomposition results.

Findings

Decomposition includes derived categories of rings with finite global dimension.

Constructs non-commutative resolutions as components of the decomposition.

Decomposition parts are Calabi-Yau and cannot be further refined.

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 $X^{ss}/G$.