On March 13th I will give a talk in the geometry seminar at the University of Oslo (UiO). Here’s the abstract: for more details see the preprint arXiv:1802.00165.

**Title: Rigidity for relative 0-cycles.**

In this talk, we will present a relation between the classical Chow group of relative 0-cycles on a regular scheme $\mathcal{X}$, projective and flat over an excellent Henselian discrete valuation ring $A$ with perfect residue field $k$, and the so-called cohomological Chow group of zero cycles of the special fiber. If $k$ is algebraically closed and with finite coefficients (prime to the residue characteristic) these groups turn out to be isomorphic. This generalizes a previous argument due to Esnault-Kerz-Wittenberg to the case of regular models with arbitrary reduction. From this, one can re-prove in case of bad reduction that the étale cycle class map for relative $0$-cycles with finite coefficients on $\mathcal{X}$ is an isomorphism, a result due to Saito and Sato in the case of semi-stable reduction. This is a joint work with Amalendu Krishna.