Luca Barbieri Viale
University of Milan, Italy
September 11, 2025
On Grothendieck’s standard conjectures: In this talk we explain the universal Weil cohomology (obtained in a recent work jointly with B. Kahn) with respect to Grothendieck’s standard conjectures (TIFR Colloquium, 1968). This new Weil cohomology is taking values in an abelian $\mathbb{Q}$-linear ($\mathbb{Q}$ is the field of rational numbers) tensor category $\mathcal{M}$. This category $\mathcal{M}$ is rigid but its $\mathbb{Q}$-algebra $E$ = End (1) of endomorphisms of the unit is not a field, a priori. (A similar picture holds true in the mixed case and both Nori motives and André’s theory of motivated cycles can be recovered as localisations.) This theory also yields a universal homological equivalence hum and a canonical comparison faithful tensor functor from Grothendieck motives (modulo hum) to $\mathcal{M}$. Standard conjectures can be translated by saying that this comparison functor is an equivalence so that $\mathrm{hum} = \mathrm{num}$ is numerical equivalence and $E = \mathbb{Q}$. A standard weaker hypothesis is then that this absolutely flat $\mathbb{Q}$-algebra $E$ is a domain hence a field: this hypothesis is equivalent to the property that $\mathcal{M}$ is Tannakian.
University of Milan, Italy
September 11, 2025
On Grothendieck’s standard conjectures: In this talk we explain the universal Weil cohomology (obtained in a recent work jointly with B. Kahn) with respect to Grothendieck’s standard conjectures (TIFR Colloquium, 1968). This new Weil cohomology is taking values in an abelian $\mathbb{Q}$-linear ($\mathbb{Q}$ is the field of rational numbers) tensor category $\mathcal{M}$. This category $\mathcal{M}$ is rigid but its $\mathbb{Q}$-algebra $E$ = End (1) of endomorphisms of the unit is not a field, a priori. (A similar picture holds true in the mixed case and both Nori motives and André’s theory of motivated cycles can be recovered as localisations.) This theory also yields a universal homological equivalence hum and a canonical comparison faithful tensor functor from Grothendieck motives (modulo hum) to $\mathcal{M}$. Standard conjectures can be translated by saying that this comparison functor is an equivalence so that $\mathrm{hum} = \mathrm{num}$ is numerical equivalence and $E = \mathbb{Q}$. A standard weaker hypothesis is then that this absolutely flat $\mathbb{Q}$-algebra $E$ is a domain hence a field: this hypothesis is equivalent to the property that $\mathcal{M}$ is Tannakian.