Hochschild cohomology of abelian categories and ringed spaces |
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Authors: | Wendy Lowen |
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Institution: | a Departement DWIS, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium b Departement WNI, Limburgs Universitair Centrum, Universitaire Campus, Building D, 3590 Diepenbeek, Belgium |
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Abstract: | This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild cohomology coincides with the one defined by Gerstenhaber, Schack and Swan in the case of module categories over diagrams and schemes and also with the Hochschild cohomology for exact categories introduced recently by Keller. In addition we show in complete generality that Hochschild cohomology satisfies a Mayer-Vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf. |
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Keywords: | primary 13D10 14A22 18E15 |
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