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Tilting objects in singularity categories and levelled mutations. arXiv:2004.02655

Preprint, arXiv:2004.02655 [math.RT] (2020).
Summary: We show the existence of tilting objects in the singularity category \(\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)\) associated to certain noetherian AS-regular algebras \(A\) and idempotents \(e\). This gives a triangle equivalence between \(\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)\) and the derived category of a finite-dimensional algebra. In particular, we obtain a tilting object if the Beilinson algebra of \(A\) is a levelled Koszul algebra. This generalises the existence of a tilting object in \(\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(S^G)\), where \(S\) is a Koszul AS-regular algebra and \(G\) is a finite group acting on \(S\), found by Iyama-Takahashi and Mori-Ueyama. Our method involves the use of Orlov’s embedding of \(\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)\) into \(\mathsf{D}^{\operatorname{b}}(\mathsf{qgr} eAe)\), the bounded derived category of graded tails, and of levelled mutations on a tilting object of \(\mathsf{D}^{\operatorname{b}}(\mathsf{qgr} eAe)\).

MSC:

16S38 Rings arising from noncommutative algebraic geometry
16G50 Cohen-Macaulay modules in associative algebras
18E30 Derived categories, triangulated categories (MSC2010)
16S35 Twisted and skew group rings, crossed products
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