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Non-commutative deformations and quasi-coherent modules

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Abstract

We identify a class of quasi-compact semi-separated (qcss) twisted presheaves of algebras \(\mathcal {A}\) for which well-behaved Grothendieck abelian categories of quasi-coherent modules \(\mathsf {Qch} (\mathcal {A})\) are defined. This class is stable under algebraic deformation, giving rise to a 1–1 correspondence between algebraic deformations of \(\mathcal {A}\) and abelian deformations of \(\mathsf {Qch} (\mathcal {A})\). For a qcss presheaf \(\mathcal {A}\), we use the Gerstenhaber–Schack (GS) complex to explicitly parameterize the first-order deformations. For a twisted presheaf \(\mathcal {A}\) with central twists, we descibe an alternative category \(\mathsf {QPr} (\mathcal {A})\) of quasi-coherent presheaves which is equivalent to \(\mathsf {Qch} (\mathcal {A})\), leading to an alternative, equivalent association of abelian deformations to GS cocycles of qcss presheaves of commutative algebras. Our construction applies to the restriction \(\mathcal {O}\) of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have \(\mathsf {Qch} (\mathcal {O}) \cong \mathsf {Qch} (X)\)). Under a natural identification of GS cohomology of \(\mathcal {O}\) and Hochschild cohomology of X, our construction is shown to be equivalent to Toda’s construction from Toda (J Differ Geom 81(1):197–224, 2009) in the smooth case.

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Acknowledgments

The authors are greatly indebted to Michel Van den Bergh for the idea of using prestacks in order to capture abelian deformations.

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Authors and Affiliations

  1. Departement Wiskunde-Informatica, Universiteit Antwerpen, Middelheimcampus, Middelheimlaan 1, 2020, Antwerp, Belgium

    Hoang Dinh Van & Wendy Lowen

  2. School of Mathematical Science, Yangzhou University, No. 180 Siwangting Road, 225002, Yangzhou, Jiangsu, China

    Liyu Liu

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  1. Hoang Dinh Van
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  2. Liyu Liu
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Correspondence to Wendy Lowen.

Additional information

The authors acknowledge the support of the European Union for the ERC Grant No. 257004-HHNcdMir and the support of the Research Foundation Flanders (FWO) under Grant No. G.0112.13N. L. L. also acknowledges the support of the Natural Science Foundation of China No. 11501492, the Natural Science Foundation of Jiangsu Province No. BK20150435 and the Natural Science Foundation for Universities in Jiangsu Province No. 15KJB110022.

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Dinh Van, H., Liu, L. & Lowen, W. Non-commutative deformations and quasi-coherent modules. Sel. Math. New Ser. 23, 1061–1119 (2017). https://doi.org/10.1007/s00029-016-0263-9

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