Hochschild Cohomology with Support
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Publication:5264495
DOI10.1093/IMRN/RNU079zbMATH Open1360.16008arXiv1304.7466OpenAlexW2063551874MaRDI QIDQ5264495
Publication date: 27 July 2015
Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)
Abstract: In this paper we investigate the functoriality properties of map-graded Hochschild complexes. We show that the category MAP of map-graded categories is naturally a stack over the category of small categories endowed with a certain Grothendieck topology of 3-covers. For a related topology of infinity-covers on the cartesian morphisms in MAP, we prove that taking map-graded Hochschild complexes defines a sheaf. From the functoriality related to "injections" between map-graded categories, we obtain Hochschild complexes "with support". We revisit Keller's arrow category argument from this perspective, and introduce and investigate a general Grothendieck construction which encompasses both the map-graded categories associated to presheaves of algebras and certain generalized arrow categories, which together constitute a pair of complementary tools for deconstructing Hochschild complexes.
Full work available at URL: https://arxiv.org/abs/1304.7466
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