Universal Weil Cohomology versus Grothendieck’s Standard Conjectures

par Luca Barbieri Viale (Université de Milan & IHES)

jeudi 5 déc. 2024, 11:00 → 12:30 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Séminaire de géométrie arithmétique

The universal Weil cohomology (obtained in a recent work jointly with B. Kahn) is taking values in an abelian Q-linear (Q is the field of rational numbers) tensor category M which is rigid but its Q-algebra E = End (1) of endomorphisms of the unit is not a field, a priori. André’s theory of motivated cycles M^A, in characteristic zero, can be recovered via the universal Weil cohomology as a localisation of M; thus E = Q  if and only if M = M^A is André’s category which is then universal for all Weil cohomologies taking values in abelian Q-linear rigid tensor categories. A similar picture holds true for the universal mixed Weil cohomology with values in MM with respect to Nori motives NM.

However, in any characteristic, this new Weil cohomology yields a universal homological equivalence hum and a canonical comparison faithful tensor functor F from Grothendieck motives (modulo hum) M^G to M. This F is an equivalence if and only if M^G is abelian, F is exact and the Grothendieck Lefschetz standard conjecture holds true. Moreover, F is an equivalence with M semi-simple if and only if hum is numerical equivalence. Therefore Grothendieck’s standard conjectures for the universal Weil cohomology or the stronger Voevodsky’s nilpotence conjecture (independently of any Weil cohomology) imply that E = Q.

 A standard hypothesis is then that this absolutely flat Q-algebra E is a domain hence a field. This hypothesis is equivalent to the property that M is Tannakian. Similarly, for MM. Note that, for every self correspondence, the trace and the Lefschetz number (as well as the coefficients of the characteristic polynomials) are defined over E. As a consequence, if E is a field all these are the same independently of the Weil cohomology.

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