Filtrations in hyperhomology
type: Item
instance of: scholarly article
title: Filtrations in hyperhomology
zbMATH Open document ID: 0913.18009
DOI: 10.1007/BF02354992
author: E. G. Sklyarenko
published in: Mathematical Notes
publication date: 1999-06-08T00:00:00Z
review text: For typical complexes, hyperhomology and its two filtrations an intrinsic description is given independent of the hyperhomology apparatus. Let \({\mathcal N}_*\) be a complex and \({\mathcal I}^*{\mathcal N}_*\) an injective Cartan-Eilenberg resolution. The filtrations of the hyperhomology of \({\mathcal N}_*\) for a functor \(F\) are determined by two filtrations of the bigraded object \({\mathcal I}^* {\mathcal N}_*\), viz., by subobjects \(K_p\) of the form \(\oplus_{i,j} {\mathcal I}^i {\mathcal N}_j\) with \(j\leq p\) (which is the first spectral sequence in the terminology of \textit{H. Cartan} and \textit{S. Eilenberg} [``Homological algebra'', Princeton Univ. Press, Princeton (1956; Zbl 0075.24305), Chap. XVII, \S 2]) and similar subobjects \(K^q\) with \(q\leq i\), the second sequence in the op. cit. Then there exists the complex \(F({\mathcal I}^*{\mathcal N}_*)\) with the standard total grading \(\oplus_{j-i=n} F({\mathcal I}^i {\mathcal N}_j)\) with respect to the total differential, \(d=d' +(-1)^j d''\), where \(d'\) and \(d''\) are differentials induced by \({\mathcal N}_*\) and \({\mathcal I}^*\) respectively. Let \(\Phi_p \mathbb{H}_m\) and \(\Phi^q \mathbb{H}_m\) be the images of the homology of its subcomplexes \(F(K_p)\) and \(F(K^q) \), respectively, in the hyperhomology \(H_m ({\mathcal N}_*;F) =H_m(F( {\mathcal I}^* {\mathcal N}_*))\) of the complex \({\mathcal N}_*\) for the functor \(F\). They induce the filtrations \[ \mathbb{H}_m({\mathcal N}_*;F) \dots \supset \Phi_{p+1} \mathbb{H}_m \supset \Phi_p \mathbb{H}_m \supset \cdots \supset \Phi_m \mathbb{H}_m \supset \Phi_{p-1} \mathbb{H}_m=0 \] and \[ \mathbb{H}_m ({\mathcal N}_*; F)= \Phi^0 \mathbb{H}_m \supset \Phi^1 \mathbb{H}_m \supset\cdots \supset \Phi^q \mathbb{H}_m \supset \dots \] in the groups \(\mathbb{H}_m ({\mathcal N}_*;F)\). Any \(F\)-acyclic Cartan-Eilenberg resolution \(C^*{\mathcal N}_*\) of the complex \({\mathcal N}_*\) admits a resolution transformation \(\alpha: C^*{\mathc…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 18G40
Mathematics Subject Classification ID: 55T05
zbMATH DE Number: 1237998
zbMATH Keywords: hyperhomology
zbMATH Keywords: Cartan-Eilenberg resolution
zbMATH Keywords: filtrations
zbMATH Keywords: spectral sequence
MaRDI profile type: Publication
cites work: Q5541099
cites work: Q3247612
cites work: Q5813687
cites work: Sur quelques points d'algèbre homologique
cites work: ON ZEEMAN'S FILTRATION IN HOMOLOGY
cites work: Q3284638
cites work: Hyper(co)homology for exact left covariant functors and a homology theory for topological spaces
description: scientific article; zbMATH DE number 1237998