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 \({{{\mathcal {D}}}}(X^{ss}{/}G)\) 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 \({{{\mathcal {D}}}}(X^{ss}/G)\).
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Notes
Variety here means an integral separated noetherian k-scheme.
The indecomposability of (twisted) NCCR’s seems to be well known to experts. It may be easily proved in a similar way as [23, Lemma A.4].
Such a K-flat resolution is constructed in the same way as for DG-algebras (see [12, Theorem 3.1.b]). One starts from the observation that for every \(M\in D(\Lambda )\) there is a morphism \(\bigoplus _{i\in I} j_{i!}(\Lambda {\mid } U_i)\rightarrow M\) with open immersions \((j_i:U_i\rightarrow X/\!\!/G)_{i\in I}\), which is an epimorphism on the level of cohomology.
Replacing \({\text {Hom}}_{X/G}(P_{G,{{{\mathcal {L}}}}},P_{G,\chi })\) with its projective resolution over \({\text {End}}_{X/G}(P_{G,{{{\mathcal {L}}}}})\).
Here we use that X is affine to identify global and local \({\text {Hom}}\)’s.
If the \(G_m\)-action is denoted by \(\gamma \) then this condition may also be written as \(Y=X^\gamma \), \(X=X^{\gamma ,-}\).
The proof follows from the Koszul resolution of \(k[X^{\lambda ,+}]\) and Bott’s theorem.
We emphasize that points in \(X^{{\mathbf {s}}}\) have trivial stabilizer. The same notation is sometimes reserved for points with closed orbit and finite stabilizer.
That is a genericity type condition that we require on \(\varepsilon \). In particular, we do not require here that \(\varepsilon \) in (weakly) generic.
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Acknowledgements
The second author thanks Jørgen Vold Rennemo and Ed Segal for interesting discussions regarding this paper. The authors thank Agnieszka Bodzenta and Alexey Bondal for their interest in this work and for useful comments on the first version of this paper. They also thank the referees for pointing out many typos and for the suggestions which improved the exposition of the paper.
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Š. Špenko is a FWO [PEGASUS\(]^2\) Marie Skłodowska-Curie fellow at the Free University of Brussels (funded by the European Union Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665501 with the Research Foundation Flanders (FWO)). During part of this work she was also a postdoc with Sue Sierra at the University of Edinburgh. Partly she was supported by L’Oréal-UNESCO scholarship “For women in science”. M. Van den Bergh is a senior researcher at the Research Foundation Flanders (FWO). While working on this project he was supported by the FWO Grant G0D8616N: “Hochschild cohomology and deformation theory of triangulated categories”. Substantial progress on this project was made during visits of the authors to each other’s host institutions. They respectively thank the University of Hasselt and the University of Edinburgh for their hospitality and support.
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Špenko, Š., Van den Bergh, M. Semi-orthogonal decompositions of GIT quotient stacks. Sel. Math. New Ser. 27, 16 (2021). https://doi.org/10.1007/s00029-021-00628-3
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DOI: https://doi.org/10.1007/s00029-021-00628-3