The relative dihedral homology of involutive algebras
type: Item
instance of: scholarly article
title: The relative dihedral homology of involutive algebras
zbMATH Open document ID: 0984.16012
DOI: 10.1155/S0161171299228074
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2000-11-13T00:00:00Z
full work available at URL: https://eudml.org/doc/49151
review text: Given \(A\), \(B\) involutive algebras and \(f\colon A\to B\) a homomorphism, the author defines a free involutive algebra resolution \(R\) of the algebra \(B\) over \(f=\pi i\), \(i\colon A\to R\) the inclusion, \(\pi\colon R\to B\) a quasiisomorphism and then considers the relative dihedral homology as: \[ {^\varepsilon HD_*}(A@>f>>B)=H_*\bigl(\tfrac R{A+[R,R]+\text{Im}(1-r^\varepsilon)}\bigr), \] where \([R,R]\) is the commutant of the algebra \(R\), \(r^\varepsilon\) is the involution on \(R\). Then he studies its main properties: (i) For \(A\) an involutive algebra \({^\varepsilon H}D_i(A\to 0)={^\varepsilon H}D_{i-1}(A)\). (ii) \({^\varepsilon H}D_i(A@>f>>B)\) does not depend on the choice of the resolution. (iii) For \(A\), \(B\) and \(C\) involutive algebras the sequence \(A@>f>>B@>g>>C\) induces a long exact sequence of relative dihedral homology: \[ \cdots\to{^\varepsilon H}D_i(A@>f>>B)\to{^\varepsilon H}D_i(A@>gf>>C)\to{^\varepsilon H}D_i(B@>g>>C)\to{^\varepsilon H}D_{i-1}(A\to B)\to\cdots\;. \]
reviewed by: Corina Mohorianu
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16E45
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 19D55
zbMATH DE Number: 1417979
zbMATH Keywords: free involutive algebras
zbMATH Keywords: resolutions
zbMATH Keywords: relative dihedral homology
zbMATH Keywords: involutions
author: Y. Gh. Gouda
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1417979