If G is a reductive group acting on a linearized smooth scheme X then we show that under suitable standard conditions the derived category \({{{\mathcal {D}}}}(X^{ss}{/}G)\) of the corresponding GIT quotient stack \(X^{ss}{/}G\) 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 Donovan–Segal that assert the existence of a semi-orthogonal decomposition of \({{{\mathcal {D}}}}(X/G)\) in which one of the parts is \({{{\mathcal {D}}}}(X^{ss}/G)\).

如果G是作用在线性平滑方案X上的还原性基团， 那么我们证明在适当的标准条件下，派生类别\（{{{\ mathcal {D}}}}（X ^ {ss} {/} G）\） GIT商堆栈\（X ^ {ss} {/} G \）的半正交分解由\（X ^ {ss} {/ \！\！/} G \）在局部具有有限的整体尺寸。分解的组成部分之一是\（X ^ {ss} {/ \！\！/} G \）的某个非交换分辨率。由作者较早地构建。作为一个具体的例子，在奇数Pfaffians的情况下，我们获得了相应商栈的半正交分解，其中所有部分都是较低或相等等级的Pfaffians的某些特定的非交换新近分辨率，这些分辨率也较早地由Pfaffians构造。作者。特别是，这种半正交分解无法进一步完善，因为它的一部分是Calabi–Yau。本文的结果补充了Halpern–Leistner，Ballard–Favero–Katzarkov和Donovan–Segal的结果，它们断言\（{{{\ mathcal {D}}}}（X / G）的半正交分解的存在\）其中一部分是\（{{{\ mathcal {D}}}}}（X ^ {ss} / G）\）。