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Modules over motivic cohomology. (English) Zbl 1180.14015
In analogy to the stable homotopy category in Topology, the introduction of the motivic stable homotopy category by Voevodsky provided a very convenient framework for defining and understanding cohomology theories for algebraic varieties. For example the motivic Eilenberg-MacLane spectrum represents motivic cohomology and is hence of fundamental interest in motivic homotopy theory. Furthermore, as in the classical situation, there are highly structured models for the motivic stable homotopy category such as the category of motivic symmetric spectra. In light of such a model the Eilenberg-MacLane spectrum \(M\mathbb{Z}\) becomes an even more interesting object. The main theorem of the present paper states that over a field \(k\) of characteristic zero, the homotopy category of \(M\mathbb{Z}\)-modules in the category of motivic symmetric spectra is equivalent to Voevodsky’s big category of motives over \(k\). In fact this holds for more general objects than \(M\mathbb{Z}\) and more generally over a noetherian and separated base scheme of finite Krull dimension for which the motivic stable homotopy category is generated by a dualizable object. One of the key point in the proof is the dualizability of varieties. This is also the moment where the assumption of characteristic zero enters the stage. Over a field \(k\) of characteristic zero every smooth quasi-projective scheme is dualizable in the motivic stable homotopy category over \(k\). To prove this generalization of a result of P. Hu [Topology 44, No. 3, 609–640 (2005; Zbl 1078.14025)] the authors use resolution of singularities. This hypotheses can be removed if one is interested in rational coefficients by using de Jong’s theorem on alterations [Publ. Math., Inst. Hautes Étud. Sci. 83, 51–93 (1996; Zbl 0916.14005)] as the authors show in the last section. But before explaining the case of rational coefficients, the authors prove the main theorem by a zig-zag of Quillen equivalences to relate Voevodsky’s motives to \(M\mathbb{Z}\)-modules. They discuss various model structures on different categories and show that their corresponding homotopy categories are isomorphic to the motivic stable homotopy category. This provides a very interesting and complete account on the properties of motivic spaces, motivic (symmetric) spectra, motivic spectra of chain complexes and their analogs with transfers. Finally, the authors also show that the category of mixed Tate motives over a noetherian and separated base scheme of finite Krull dimension is equivalent to the associated homotopy category of the module category of a differential graded algebra with many objects.

14F42 Motivic cohomology; motivic homotopy theory
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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