Hochschild cohomology

General Introduction
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Search results
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Online References
Brief intro in Caldararu's notes on Derived categories of sheaves, in Homol alg folder or on arXiv.
Schuhmacher: Hochschild cohomology of complex spaces and noetherian schemes, HHA vol 6
Yekutieli: The continuous Hochschild cochain complex of a scheme, Can J Math Vol 54 (6).
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Paper References
Short exposition of Hochschild cohomology (for associative algebras) in Ch 11 of Pierce (el), under Algebras.
For cohomology of algebras, see CartanEilenberg, and maybe MacLane: Homology.
Course notes from Grojnowski's lectures in Cambridge.
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Definition
This can be regarded as an instance of Monad cohomology.
For Hochschild cohomology of Segal cats and model cats, see Toen: Homotopical and higher categorical structures in algebraic geometry. File Toen web unpubl hab.pdf. Idea: Hochschild cohomology should be the space of endomorphisms of an identity functor.
For Hochschild cohomology of DGcategories, see Toen: Lecture on DGcategories. File Toen web unpubl swisk.pdf. Treats basic theory, localization, relation to model cats, functorial cones, Ktheory and Hochschild cohomology, and descent problems.
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
arXiv:1102.5756 Cohomology of exact categories and (non)additive sheaves from arXiv Front: math.KT by Dmitry Kaledin, Wendy Lowen We use (non)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Ext's in a suitable bisheaf category. We compare our approach to various definitions present in the literature.
Various things by Toen on Hochschild stuff in abstract settings.
arXiv:1001.5379 The homology of digraphs as a generalisation of Hochschild homology from arXiv Front: math.KT by Paul Turner, Emmanuel Wagner J. Przytycki has established a connection between the Hochschild homology of an algebra $A$ and the chromatic graph homology of a polygon graph with coefficients in $A$. In general the chromatic graph homology is not defined in the case where the coefficient ring is a noncommutative algebra. In this paper we define a new homology theory for directed graphs which takes coefficients in an arbitrary $AA$ bimodule, for $A$ possibly noncommutative, which on polygons agrees with Hochschild homology through a range of dimensions.
Toen: Algebres simplicicales etc, file Toen web prepr rhamloop.pdf. Comparison between functions on derived loop spaces and de Rham theory. Take a smooth kalgebra, k aof char zero. Then (roughly) the de Rham algebra of A and the simplical algebra determine each other (functorial equivalence). Consequence: For a smooth kscheme , the algebraic de Rham cohomology is identified with equivariant functions on the derived loop space of . Conjecturally this should follow from a more general comparison between functions on the derived loop space and cyclic homology. Also functorial and multiplicative versions of HKR type thms on decompositions of Hochschild cohomology, for any separated kscheme.
Title: Operads of natural operations I: Lattice paths, braces and Hochschild cochains. Authors: Michael Batanin, Clemens Berger and Martin Markl. In this first paper of a series we study various operads of natural operations on Hochschild cochains and relationships between them. http://arxiv.org/abs/0906.4097
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Other Information
http://mathoverflow.net/questions/28472/bookonhochschildcohomology
http://mathoverflow.net/questions/33877/hochschildcohomologyofaandofmoda
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