Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, by Ayoub Joseph

This is our PhD thesis (under the supervision of F. Morel). In SGA 4 and SGA 7, Grothendieck et alii developed a powerful machinery to study étale cohomology of algebraic varieties. This machinery is based on seven operations: f^*, f_*, f_!, f^!, tensor product, internal homs and the nearby cycles \Psi, together with several theorems. In this thesis we develop the analogue machinery in the motivic context. We rely on work of Jardine, Morel and Voevodsky that gives the operations f^*, f_*, tensor product and internal homs for the motivic stable homotopy categories SH(-). We construct out of these operations the remaining ones: f_!, f^! and \Psi. We then extend many of the fundamental results in the étale setting to the motivic one: the base change theorems, the constructibility of the seven operations, their exactitude with respect to the homotopy t-structure, the duality formulas, the computation of the nearby cycles, etc.

P.S. The first chapter of this thesis was already posted on the K-theory server in January 7, 2005, as preprint 717.

[ The original version of this thesis, posted December 16, 2005, has been updated on June 9, 2006. ]

Ayoub Joseph <ayoub@math.jussieu.fr>