Chain complexes of symmetric categorical groups
type: Item
instance of: scholarly article
title: Chain complexes of symmetric categorical groups
zbMATH Open document ID: 1068.18015
author: Xianqiang Yang
published in: Journal of Pure and Applied Algebra
publication date: 2005-04-07T00:00:00Z
review text: The authors revisit previous results using recent techniques developed in higher-dimensional homological algebra. The Hattori-Villamayor-Zelinsky long exact sequence connecting Amitsur cohomology groups of a commutative algebra with coefficients \(\mathcal U\) (the group of units) and \({\mathcal Pic}\) (the Picard Group) and the Ulbrich cohomology are derived as special instances of general results on the symmetric categorical groups. The kernel and the cokernel of a morphism between symmetric categorical groups previously introduced are refined here by the notions of kernel and cokernel relative to a natural transformation. Using these, the authors define the cohomology categorical groups of a complex of symmetric categorical groups. Two definitions are investigated as in the case of abelian groups, giving equivalent categorical groups. Extensions of categorical groups are introduced and a long 2-exact sequence of cohomology categorical groups is associated to any extension of complexes of symmetric categorical groups. As a special case of this sequence, the authors obtain the Ulbrich cohomology, the Hattori-Villamayor-Zelinsky exact sequences and a special case of the kernel-cokernel lemma for symmetric categorical groups. Simplicial cohomology with coefficients in a symmetric categorical group is also discussed.
reviewed by: Constantin Năstăsescu
Mathematics Subject Classification ID: 18G50
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 13D99
Mathematics Subject Classification ID: 18G30
zbMATH DE Number: 2153397
zbMATH Keywords: symmetric categorical group
zbMATH Keywords: higher dimensional homological algebra
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.08.029
OpenAlex ID: W2149309724
cites work: Extensions of symmetric cat-groups
cites work: Brauer groups, Hopf algebras and Galois theory
cites work: (Braided) tensor structures on homotopy groupoids and nerves of (Braided) categorical groups1
cites work: Simplicial cohomology with coefficients in symmetric categorical groups
cites work: Homotopy classification of categorial torsors
cites work: Q4663632
cites work: Q4453907
cites work: Q3156496
cites work: On groups \(H^n(S/R)\) related to the Amitsur cohomology and the Brauer group of commutative rings
cites work: Q4948681
cites work: Monoidal bicategories and extensions of gr-categories
cites work: On Villamayor and Zelinsky’s long exact sequence
cites work: The mapping cone method and the Hattori-Villamayor-Zelinsky sequences
cites work: Complexes of categories with Abelian group structure
cites work: Q3964690
cites work: Group cohomology for Picard groups
cites work: Brauer groups and Amitsur cohomology for general commutative ring extensions
cites work: A Picard-Brauer exact sequence of categorical groups
DOI: 10.1016/J.JPAA.2004.08.029
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description: scientific article; zbMATH DE number 2153397