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 Astérisque 2007; 362 pp; softcover Number: 315 ISBN-10: 2-85629-245-3 ISBN-13: 978-2-85629-245-7 List Price: US$117 Individual Members: US$105.30 Order Code: AST/315 This second volume contains chapters 3 and 4 of the author's study of the functoriality of the stable homotopy categories of schemes. In the previous volume, he concentrated on the six operations $$f^*$$, $$f_*$$, $$f_!$$, $$f^!$$, $$-\otimes -$$ and $$\underline{\mathbf {Hom}} (-,-)$$, their constructibility and exactness. This volume begins with the construction of the nearby motive functors $$\Psi _f$$ which are the analogue of the nearby cycles functors, well-known in étale cohomology. The author then extends the vanishing cycles formalism to these functors. In particular, he computes the effect of the functor $$\Psi _f$$ in the case where $$f$$ has semi-stable reduction. He also shows that $$\Psi _f$$ preserves constructible motives and commutes with external tensor product and duality. He then defines a monodromy operator and proves that this operator is nilpotent. The last chapter, which is of a different nature than the previous ones, recalls in full detail the construction of the stable homotopy category of $$S$$-schemes. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in number theory. Table of Contents La théorie des foncteurs cycles proches dans un cadre motivique La construction de 2-foncteurs homotopiques stables Bibliographie