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 Astérisque 2007; 464 pp; softcover Number: 314 ISBN-10: 2-85629-244-5 ISBN-13: 978-2-85629-244-0 List Price: US$132 Individual Members: US$118.80 Order Code: AST/314 By the work of Morel, Voevodsky, and other mathematicians, one has the notion of the stable motivic homotopy type of a smooth $$S$$-scheme. This object lives in the stable homotopy category of $$S$$-schemes SH$$(S)$$. This work consists of two volumes and each of them is divided into two chapters. In the first chapter, the author shows that from the viewpoint of functoriality, the categories SH$$(S)$$ behave like the derived categories of $$l$$-adic sheaves. Indeed, the formalism of Grothendieck operations $$f^*$$, $$f_*$$, $$f_!$$ and $$f^!$$ extends to the motivic world. In the second chapter, the author studies the constructibility of motives and develops Verdier duality. The third chapter deals with the theory of nearby motives and vanishing motives. The last chapter provides a self-contained treatment of the construction of the categories SH$$(S)$$. 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 algebra and algebraic geometry. Table of Contents Les quatre opérations de Grothendieck dans un cadre motivique Compléments sur les 2-foncteurs homotopiques stables et les quatre opérations Bibliographie