Bloch's cycle complex and coherent dualizing complexes in positive characteristic

Abstract

Let $X$ be a separated scheme of dimension $d$ of finite type over a perfect field $k$ of positive characteristic $p$. In this work, we show that Bloch's cycle complex ${\mathbb{Z}}_X^c$ of zero cycles mod $p^n$ is quasi-isomorphic to the Cartier operator fixed part of a certain dualizing complex from coherent duality theory. From this we obtain new vanishing results for the higher Chow groups of zero cycles with mod $p^n$ coefficients for singular varieties.

arXiv comment62 pages. Accepted for publication in Tran AMS. Compared to v2: Due to an update of arXiv:1703.02269 concerning the normality condition (more specifically, Thm. 8.6), Cor. 8.12 in our article is now changed. (The statement in Cor. 0.2(6) remains valid.)