# Codimension two cycles and unramified third cohomology for certain products of two varieties – Jean-Louis Colliot-Thelene (Université de Paris-Saclay)

#### Venue

Dipartimento di Matematica, Aula Magna.

#### Abstract

We investigate the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field, under the assumption that the surface is geometrically $\mathsf{CH}_0$-trivial. By this we mean that over any algebraically closed field extension, the degree map on zero-dimensional Chow group of the surface is an isomorphism. This applies to Enriques surfaces. When the Néron-Severi group has no torsion, we recover earlier results of A. Pirutka. This is joint work with Federico Scavia.

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