Baptiste Morin

On the Weil-étale cohomology of number fields

We give a direct description of the category of sheaves on Lichtenbaums Weil- etale site of a number ring. Then we apply this result to define a spectral sequence relating Weil-etale cohomology to Artin-Verdier etale cohomology. Finally we construct complexes of etale sheaves computing the expected Weil-etale cohomology.

The Weil-étale fundamental group of a number field II

We define the fundamental group underlying the Weil-étale cohomology of number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced by Lichtenbaum (Ann Math 170(2):657–683, 2009). We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents first degree cohomology with coefficients in locally compact abelian groups. We apply this result to compute the Weil-étale cohomology in low degrees and to prove that the Weil-étale topos of a number ring satisfies the expected properties of the conjectural Lichtenbaum topos.

Utilisation d’une cohomologie étale équivariante en topologie arithmétique

Sikora has given results which confirm the analogy between number fields and 3-manifolds. However, he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora’s formulas which leads us to clarify and to extend the dictionary of arithmetic topology.

The classifying topos of a group scheme and invariants of symmetric bundles

Let

On the trace form of Galois algebras: ON THE TRACE FORM OF GALOIS ALGEBRAS

Y

On the Weil-Étale Topos of Regular Arithmetic Schemes

be a scheme in which 2 is invertible and let

Sur le topos Weil-étale d'un corps de nombres

V

Hopf algebras and quadratic forms

be a rank

THE WEIL-ÉTALE FUNDAMENTAL GROUP OF A NUMBER FIELD I

n

Weil-\'etale cohomology and Zeta-values of proper regular arithmetic schemes

vector bundle on