Mixed motives and their realization in derived categories
type: Item
instance of: scholarly article
title: Mixed motives and their realization in derived categories
zbMATH Open document ID: 0938.14008
published in: Lecture Notes in Mathematics
publication date: 1995-08-10T00:00:00Z
review text: The main goal of the still conjectural theory of motives is to construct a category of motivic sheaves \({\mathcal M}{\mathcal M}(V)\) for any algebraic variety \(V\). This construction should carry the standard six functorial operations introduced by Grothendieck into any cohomology theory. Also it should be supplied by a derived category with heart \({\mathcal M}{\mathcal M}(V)\) and a canonical filtration by pure motives of some weight. This construction must lead to two kinds of cohomology theories: geometric cohomologies and absolute motivic cohomologies connected by a spectral sequence. The values of the geometric cohomologies belong to the category \({\mathcal M}{\mathcal M}(\text{Spec}(k))\) of motivic sheaves over a point and the values for the absolute theory lie in abelian groups. Any known cohomology theory can be considered as a realization (some kind of truncation) of this universal construction. Among the well established realizations we can see étale cohomologies and Saito-Hodge cohomologies. In particular, for the latter theory geometric cohomologies are the De Rham theory (with their mixed Hodge structure) and absolute cohomologies will be the Deligne cohomologies. \textit{U. Jannsen} [``Mixed motives and algebraic K-theory'', Lect. Notes Math. 1400 (1990; Zbl 0691.14001)] has introduced some category \({\mathcal M}{\mathcal R}(V)\) served as an approximation to the category \({\mathcal M}{\mathcal M}(V)\) and constructed the geometric cohomologies in this case. The author returns back to Jannsen's construction and defines the absolute cohomology theory in this case. She proves all the properties which the motivic people expect including the spectral sequence connecting the both theories and the Bloch-Ogus axioms for cohomologies. She goes even further defining the Chern classes with values in the new cohomologies and proving relations between the elements of higher K-groups with vanishing Chern classes and the elements of the motivic extension gro…
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 19E15
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 19-02
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 14F43
zbMATH DE Number: 783767
zbMATH Keywords: derived category
zbMATH Keywords: motivic sheaves
zbMATH Keywords: de Rham theory
zbMATH Keywords: mixed Hodge structure
zbMATH Keywords: Deligne cohomologies
zbMATH Keywords: motivic extension group
zbMATH Keywords: higher \(K\)-groups
zbMATH Keywords: absolute cohomologies
author: Annette Huber-Klawitter
reviewed by: A. N. Parshin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0095503
OpenAlex ID: W573663564
DOI: 10.1007/BFB0095503
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description: scientific article; zbMATH DE number 783767