Mixed motives and algebraic cycles. I
The idea of algebraic motives goes essentially back to A. Grothendieck. In the 1960's, when working on the Weil conjecture on zeta functions of algebraic varieties over finite fields, he postulated certain conjectures on algebraic cycle classes (Grothendieck's standard conjectures), which would imply an affirmative answer to the Weil conjecture. These standard conjectures concern general Weil cohomology theories, and they indicate the existence of a certain semi-simple abelian category \({\mathcal M}(k)\) depending on the groundfield \(k\) which is universal with respect to Weil cohomology functors, and that in the following sense: To every smooth projective variety \(X/k\), there corresponds functorially an object \(h(X)\in \text{obj} ({\mathcal M}(k))\), and any Weil cohomology functor factors through this ``motivic'' functor. Thus the functor \(h\) (of pure Grothendieck motives) is the universal Weil cohomology on smooth projective varieties, with values in a semi-simple abelian category. -- Now, in view of the progress made in the theory of mixed Hodge structures, higher Chow groups and higher \(K\)-groups in algebraic geometry, there is some hope for the existence of a bigger universal category, the category of mixed motives, which contains the motives of even singular quasiprojective varieties. In the present paper, the author outlines the construction of a triangulated category \({\mathcal D}(k)\), whose objects are called mixed motives over \(k\) and which has so far all the expected formal properties. The objects of \({\mathcal D}(k)\), that is the mixed motives, appear here as diagrams of smooth projective varieties and certain correspondences between them, where these correspondences are defined via S. Bloch's theory of higher Chow groups [cf.: \textit{M. Levine}, ``Bloch's higher Chow groups revisited'' (to appear)]. The construction of the ``motivic'' category \({\mathcal D}(k)\) as well as its first basic properties are just briefly outlined; a full account on the subject will be given in a forthcoming paper. However, this survey (and announcement) is highly interesting, also because the author lucidly discusses the link between his construction of mixed motives and the standard conjectures on algebraic cycle classes in Weil cohomologies and filtrations of Chow groups.
standard conjectures on algebraic cycle classes
scientific article; zbMATH DE number 858527