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The Grothendieck six operations and the vanishing cycles formalism in the motivic world. I. (Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I.) (French) Zbl 1146.14001
Astérisque 314. Paris: Société Mathématique de France (ISBN 978-2-85629-244-0/pbk). x, 466 p. (2007).
The complete work of the author consists of two volumes and four chapters. In the first- under review - volume two chapters are included. In the first chapter the author considers the stable homotopy category of $$S$$-schemes $${\mathbf{SH}}(S)$$ as developed by V. Voevodsky, F. Morel and others. The main point of the first chapter is to show that the Grothendieck operations $$f^{*},$$ $$f_{*},$$ $$f_{!}$$ and $$f^{!}$$ can be extended to the motivic setting. In the first introductory section the author gives preliminaries concerning $$2$$-categories. This includes the notions of $$2$$-category, $$2$$-functor, adjunctions in a $$2$$-category etc.. In the second section the exchange structures of 2-functors are studied. The cross functors are described. Recall that a cross functor from a category $$\mathcal C$$ to a $$2$$-category $$\mathcal D$$ is given by the following data: four $$2$$-functors: $$H^{*}$$, $$H_{*}$$, $$H_{!}$$, $$H^{!}$$, four exchange structures on each couple: $$(H^{*},H^{!})$$, $$(H_{*},H_{!})$$, $$(H^{*},H_{!})$$, $$(H_{*},H^{!})$$ satisfying certain axioms. Third section is devoted to the extension of 2-functors. Suppose that a category $$\mathcal C$$ and a $$2$$-category $$\mathcal D$$ are given. Assume further that we have two $$2$$-functors $$H_{1}$$ and $$H_{2}$$ defined on two subcategories $$\mathcal C_{1}$$ and $$\mathcal C_{2}.$$ The sufficient condition for the existence of a $$2$$- functor $$H$$ on the category $$\mathcal C$$, such that $$H_{1}$$ and $$H_{2}$$ come from $$H,$$ is established. This fact is used in section 6, where the $$2$$-functor $$H^{!}$$ is constructed. There $${\mathcal C}=Sch/S$$ is the category of quasi-projective $$S$$-schemes, $${\mathcal C}_{1}=(Sch/S)^{Imm}$$ is the category of quasi-projective $$S$$-schemes with closed immersions as morphisms, $${\mathcal C}_2=(Sch/S)^{Liss}$$ is the category of quasi-projective $$S$$-schemes with smooth $$S$$-morphisms as morphisms. In section 4 the $$2$$-functors $$H_{1}$$ and $$H_{2}$$ are described. In section $$5$$ the stability axiom is studied and Thom equivalence is constructed. In section $$7$$ some consequences of the stability axiom are considered. Among basic theorems in étale cohomology one has: the theorems cocerning: the costructibility of cohomology sheaves $$R^{i}f_{*}{\mathcal F}$$ for a morphism $$f$$ of finite type and $${\mathcal F}$$ a constructible sheaf of $${\Lambda}$$-modules, cohomology dimension of $$Rf_{*}$$, Verdier duality. Chapter $$2$$ is devoted to establishing motivic analogs of these.

##### MSC:
 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14C25 Algebraic cycles 14F20 Étale and other Grothendieck topologies and (co)homologies 14F35 Homotopy theory and fundamental groups in algebraic geometry 14F42 Motivic cohomology; motivic homotopy theory 18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) 18F10 Grothendieck topologies and Grothendieck topoi 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 18G55 Nonabelian homotopical algebra (MSC2010) 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects)
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