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

TL;DR

This paper establishes a connection between Bloch's cycle complex and dualizing complexes in positive characteristic, leading to new vanishing results for higher Chow groups of zero cycles on singular varieties.

Contribution

It demonstrates that Bloch's cycle complex mod $p^n$ is quasi-isomorphic to the Cartier operator fixed part of a dualizing complex, revealing a novel link in positive characteristic.

Findings

Bloch's cycle complex is quasi-isomorphic to a fixed part of a dualizing complex.

New vanishing results for higher Chow groups of zero cycles on singular varieties.

Establishes a deep connection between cycle complexes and coherent duality 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 $\mathbb{Z}^c_X$ 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.