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Triangulated categories of motives over a field. (English) Zbl 1019.14009
Voevodsky, Vladimir et al., Cycles, transfers, and motivic homology theories. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 143, 188-238 (2000).
The aim of this paper is to construct, for any perfect field $$k$$, a triangulated category $$\text{DM}^{eff}(k)$$ which is called the “triangulated category of effective motivic complexes over $$k$$”. This construction provides a natural categorical framework to study different algebraic cycle cohomology theories in a unified way, that is in the same way as the derived category of étale sheaves provides a categorical framework for investigating étale cohomology, and the final outcome is a certain rigid tensor triangulated category $$\text{DM}_{gm}^{eff}(k)$$, the so-called “triangulated category of effective geometric motives over $$k$$”, together with a complex $$\mathbb{Z}(n)$$, $$n\geq 0$$, of sheaves over it, which allows to define a motivic cohomology theory (à la A. Beilinson) over $$k$$. This motivic cohomology theory satisfies most of the axioms postulated by Beilinson and others, on the one hand, and it is closely related to many other known algebraic cycle (co-)homology theories, on the other hand.
Most of the proofs in this paper are heavily based on the results developed in the foregoing articles of this volume. In this vein, the present paper must be seen as the ultimate highlight of the whole volume.
For the entire collection see [Zbl 1021.14006].

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 18E30 Derived categories, triangulated categories (MSC2010)