Mixed motives and algebraic cycles. II
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 the other constructions higher Chow groups are revealed at a later stage of the respective theories. As for the contents, the paper under review consists of six sections entitled as follows: 1. Cycle complexes and higher Chow groups; 2. Grothendieck's construction; 3. \(C\)-complexes; 4. The category \({\mathcal D}(k)\); 5. Cohomology realizations of motives; 6. Cohomological motives of quasi-projective varieties. Sections 1--3 are of preliminary nature with regard to the construction of the triangulated category \({\mathcal D}(k)\). Among the new concepts introduced here are: distinguished subcomplexes of a cycle complex, the category of formal symbols, and \(C\)-complexes as a generalization of double complexes. The explicit definition (and construction) of the category \({\mathcal D}(k)\) is given in Section 4, and its basic structures are studied here as well. In Section 5, the cohomology realization functor is constructed for Betti cohomology. As the author points out, the construction of étale cohomology with \(\mathbb Q_\ell\)-coefficients is also possible, in this framework, but will be treated in a separate paper. Section 6 deals with the interplay between quasi-projective varieties in characteristic zero and the author's mixed motives, including the construction of versions of higher Chow (co-)homology groups with (or without) compact support. At the end of this paper, the author refers again to Part III of the series, where the conjectural properties of the (now established) category \({\mathcal D}(k)\) have already been discussed. With a view to Section 6, a broader discussion can be found in the author's more recent paper ``Homological and cohomological motives of algebraic varieties'' [Invent. Math. 142, No.~2, 319--349 (2000; Zbl 1041.14502)].
scientific article; zbMATH DE number 2132033