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The Grothendieck six operations and the vanishing cycles formalism in the motivic world. II. (Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II.) (French) Zbl 1153.14001
Astérisque 315. Paris: Société Mathématique de France (ISBN 978-2-85629-245-7/pbk). 362 p. (2007).
This second volume contains chapters 3 and 4 [cf. Astérisque 314. (Paris): Société Mathématique de France. (2007; Zbl 1146.14001)]. In chapter 3 the construction of nearby motive functors $${\Psi}_{f}$$ is given. These functors are analogous to the nearby cycle functors in étale cohomology. In section 3.1 the author introduces a notion of a system of specialization and derives some formal properties of it. Section 3.2 is devoted to the construction of new systems of specializations out of the given diagram of schemes and a system of specialization. The construction of the nearby functors is a particular case of this procedure. In section 3.3 the most important theorems concerning systems of specializations and nearby functors are established. In particular the effect of the functor $${\Psi}_{f}$$ for the case when $$f$$ has semistable reduction is computed. In section 3.4 and 3.5 two systems of specialization $$\mathcal\Gamma$$ and $${\Psi}$$ are studied. It is shown that $${\Psi}_{f}$$ preserve constructible motives and commute with external products and duality. In section 3.6 the monodromy operator is defined and it is proven that this operator is nilpotent.
Chapter 4 is devoted to the construction in full details of the homotopy category of $$S$$-schemes. It relies on the works of F. Morel and V. Voevodsky [Publ. Math., Inst. Hautes Étud. Sci. 90, 45–143 (1999; Zbl 0983.14007)], [J. F. Jardine, J. Pure Appl. Algebra 47, 35–87 (1987; Zbl 0624.18007), Doc. Math., J. DMV 5, 445–553 (2000; Zbl 0969.19004)], M. Hoovey [J. Pure Appl. Algebra 165, No. 1, 63–127 (2001; Zbl 1008.55006)] and others.

##### MSC:
 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies 14F35 Homotopy theory and fundamental groups in algebraic geometry 14F42 Motivic cohomology; motivic homotopy theory
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