(HoRSe seminar) Realizations of motives

(HoRSe seminar) Realizations of motives

Tue, 01/06/2010
15:45 		Denis-Charles Cisinski (Paris 13) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) Realizations of motives
A categorification of cycle class maps consists to define realization functors from constructible motivic sheaves to other categories of coefficients (e.g. constructible $ l $-adic sheaves), which are compatible with the six operations. Given a field $ k $, we will describe a systematic construction, which associates, to any cohomology theory $ E $, represented in $ DM(k) $, a triangulated category of constructible $ E $-modules $ D(X,E) $, for $ X $ of finite type over $ k $, endowed with a realization functor from the triangulated category of constructible motivic sheaves over $ X $. In the case $ E $ is either algebraic de Rham cohomology (with $ char(k)=0 $), or $ E $ is $ l $-adic cohomology, one recovers in this way the triangulated categories of $ D $-modules or of $ l $-adic sheaves. In the case $ E $ is rigid cohomology (with $ char(k)=p>0 $), this construction provides a nice system of $ p $-adic coefficients which is closed under the six operations.