Noncommutative Mackey functors and Hopf-cyclic homology.
type: Item
instance of: scholarly article
title: Noncommutative Mackey functors and Hopf-cyclic homology.
zbMATH Open document ID: 1123.16007
author: Jolanta Słomińska
published in: \(K\)-Theory
publication date: 2007-02-13T00:00:00Z
review text: Duality of Hopf-cyclic homology and cohomology is described by using Mackey functors. The description is based on the fact that for every natural equivalence \(\omega\colon F_1|_{\mathcal C}\to F_2|_{\mathcal C}\), \({\mathcal C}={\mathcal C}_1\cap{\mathcal C}_2\), there exist functors \(F_i'\colon{\mathcal C}'\to{\mathcal A}\), \({\mathcal C}'= {\mathcal C}_1\cup_{\mathcal C}{\mathcal C}_2\) and a natural equivalence \(\omega'\colon F_1'\to F_2'\) such that \(F_i'|_{\mathcal C_i}=F_i\) and \(\omega'|_{\mathcal C}=\omega\) (Prop.1.2). For this purpose, first the categories \({\mathcal F}(\text{Ass})\) and \({\mathcal F}_+(\text{Ass})\), whose objects are finite sets \([n]=\{0,\dots,n\}\) and finite sets and the empty set \(\{-1\}\) with orderings, and the categories \(\Delta\) and \(\Delta_+\) whose objects are the same as that of \({\mathcal F}(\text{Ass})\) and \({\mathcal F}_+(\text{Ass})\) and morphisms are compositions of injections and surjections, are introduced (\S2.2). Then, as subcategories of \({\mathcal F}(\text{Ass})\), \({\mathcal F}_\bullet(\text{Ass})\) and \({\mathcal F}^\bullet(\text{Ass})\) are defined. They have the same objects as \({\mathcal F}(\text{Ass})\), but objects are based maps \(f\colon[n]\to[m]\); \(f(0)=0\) or \(f(n)=m\). For a left (right) \(A\)-module \(M\), the functors \({\mathbf M}^\bullet\) and \({\mathbf M}_\bullet\) are defined by \[ {\mathbf M}^\bullet[n]=A^{\otimes n}\otimes_kM,\quad{\mathbf M}_\bullet[n]=M\otimes_kA^{\otimes n}. \] Let \(J\colon\Delta^{\text{op}}_+\to\Delta\) be the morphism \(J[n]= [n+1]\), then Hochschild complexes for an \(A\)-module and for a \(C\)-bicomplex, are \({\mathbf M}J|_{\Delta^{\text{op}}}\) and \({\mathbf M}J^{\text{op}}|_\Delta\) (\S2.4). The category \(\mathcal{QF}(\text{Ass})\) is a particular case of the generalized Quillen \(\mathcal Q\)-construction [\textit{B. L. Feigin} and \textit{B. L. Tsygan}, K-theory, arithmetic and geometry, Semin., Moscow Univ. 1984-86, Lect. Notes Math. 1289,…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 5125387
zbMATH Keywords: Hopf-cyclic homology
zbMATH Keywords: nonlinear Mackey functors
zbMATH Keywords: cyclic objects
zbMATH Keywords: Hopf-cyclic cohomology
zbMATH Keywords: Hopf algebras
zbMATH Keywords: anti-Yetter-Drinfeld modules
zbMATH Keywords: Hochschild complexes
MaRDI profile type: Publication
full work available at URL: s10977-006-9002-x
OpenAlex ID: W2044522789
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DOI: 10.1007/S10977-006-9002-X
description: scientific article; zbMATH DE number 5125387