Frédéric Déglise

Mixed Weil cohomologies

Abstract We define, for a regular scheme S and a given field of characteristic zero K, the notion of K-linear mixed Weil cohomology on smooth S-schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance, stability (which means that the cohomology of G m behaves correctly), and Kunneth formula. We prove that any mixed Weil cohomology defined on smooth S-schemes induces a symmetric monoidal realization of some suitable triangulated category of motives over S to the derived category of the field K. This implies a finiteness theorem and a Poincare duality theorem for such a cohomology with respect to smooth and projective S-schemes (which can be extended to smooth S-schemes when S is the spectrum of a perfect field). This formalism also provides a convenient tool to understand the comparison of such cohomology theories.

Orientable homotopy modules

We prove a conjecture of Morel identifying Voevodsky’s homotopy invariant sheaves with transfers with spectra in the stable homotopy category which are concentrated in degree zero for the homotopy

THE RIGID SYNTOMIC RING SPECTRUM

t

Modules de cycles et motifs mixtes

-structure and have a trivial action of the Hopf map. This is done by relating these two kind of objects to Rost’s cycle modules. Applications to algebraic cobordism and construction of cycle classes are given.

Triangulated categories of mixed motives

The aim of this paper is to show that Besser syntomic cohomology is representable by a rational ring spectrum in the motivic homotopical sense. In fact, extending previous constructions, we exhibit a simple representability criterion and we apply it to several cohomologies in order to get our central result. This theorem gives new results for syntomic cohomology such as h-descent and the compatibility of cycle classes with Gysin morphisms. Along the way, we prove that motivic ring spectra induces a complete Bloch-Ogus cohomological formalism and even more. Finally, following a general motivic homotopical philosophy, we exhibit a natural notion of syntomic coefficients.

Around the Gysin triangle II

For a perfect field k, we give a relation between the category of homotopy invariant sheaves with transfers defined by Voevodsky and the category of cycle modules defined by Rost. More precisely, the category of cycle modules over k is equivalent to the category obtained from the homotopy invariant sheaves with transfers by formally inverting the sheaf represented by Gm with its canonical structure of a presheaf with transfers. This gives a canonical monoidal structure on the category of cycle modules over k, and shows that it is Abelian. To cite this article: F. Deglise, C. R. Acad. Sci. Paris, Ser. I 336 (2003).