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Motives and modules over motivic cohomology. (English) Zbl 1097.14016
Summary: We summarize the main results and techniques in our homotopical algebraic approach to motives. A major part of this work relies on highly structured models for motivic stable homotopy theory. For any noetherian and separated base scheme of finite Krull dimension these frameworks give rise to a homotopy theoretic meaningful study of modules over motivic cohomology. When the base scheme is Spec($$k$$), for $$k$$ a field of characteristic zero, the corresponding homotopy category is equivalent to Voevodsky’s big category of motives.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 55P42 Stable homotopy theory, spectra 55U35 Abstract and axiomatic homotopy theory in algebraic topology 18G30 Simplicial sets; simplicial objects in a category (MSC2010) 18G55 Nonabelian homotopical algebra (MSC2010)
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