Mixed motives and algebraic cycles. II
type: Item
instance of: scholarly article
title: Mixed motives and algebraic cycles. II
zbMATH Open document ID: 1068.14022
author: Masaki Hanamura
published in: Inventiones Mathematicae
publication date: 2005-02-02T00:00:00Z
review text: This paper is the second of a series of papers on mixed motives and algebraic cycles. Part I [Math. Res. Lett. 2, No. ~6, 811--821 (1995; Zbl 0867.14003)] and Part III [Math. Res. Lett. 6, No. ~1, 61--82 (1999; Zbl 0968.14004)] have already been reviewed in detail, according to their respective particular purpose. Whilst Part I provided a survey of the author's general program of constructing the triangulated category \({\mathcal D}(k)\) of mixed motives over a field \(k\), whose existence was suggested by the fundamental work of P. Deligne, A. Beilinson, S. Bloch, U. Jannsen, M. Levine, and others, Part III was mainly devoted to the discussion of several conjectural properties of the category \({\mathcal D}(k)\) to be constructed. The intermediate Part II, which is now under review, represents the most extensive, elaborated and substantial part of the entire series under this title. Namely, in the present part, the author carries out the anticipated construction of the triangulated category \({\mathcal D}(k)\) of mixed motives in full detail and rigor, just as partly announced in Part I. Although other categorical constructions for mixed motives have been published, in the meantime, most importantly those proposed by \textit{M. Levine} [Mixed Motives (Math. Surv. Monographs 57, AMS, Providence, RI)(1998; Zbl 0902.14003)] and \textit{V. Voevodsky} [in: Cycles, transfers, and motivic cohomology theories (Princeton Univ. Press, Princeton, N. J. , Ann. Math. Stud. 143, 188--238 (2000; Zbl 1019.14009)), the approach presented here must be seen as being highly original, fundamental and interesting, all the more as it has been conceptually designed (and developed) over a period of more than a decade. The author perceives a mixed motive as a diagram built up of smooth projective varieties and correspondences (in the sense of higher Chow groups) between them. In this special conception, higher correspondences are naturally built into the very definition of motives, while in…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 19E15
Mathematics Subject Classification ID: 14C35
zbMATH DE Number: 2132033
zbMATH Keywords: triangulated categories
zbMATH Keywords: quasi-projective varieties
zbMATH Keywords: motivic cohomology
zbMATH Keywords: Chow groups
zbMATH Keywords: \(K\)-theory
MaRDI profile type: Publication
full work available at URL: s00222-004-0369-0
OpenAlex ID: W2032570853
DOI: 10.1007/S00222-004-0369-0
description: scientific article; zbMATH DE number 2132033