A moving lemma for relative $0$-cycles

TL;DR

This paper establishes a moving lemma for relative 0-cycles on certain schemes, enabling better algebraic cycle representations and linking crystalline cohomology to algebraic cycles.

Contribution

It introduces a new moving lemma for higher Chow groups of relative 0-cycles, facilitating algebraic cycle representations with desirable properties.

Findings

Cycle classes can be represented by cycles with finiteness, surjectivity, and smoothness.

Crystalline cohomology of smooth varieties can be expressed via algebraic cycles.

Provides tools for algebraic cycle manipulations in algebraic geometry.

Abstract

We prove a moving lemma for the additive and ordinary higher Chow groups of relative $0$-cycles of regular semi-local $k$-schemes essentially of finite type over an infinite perfect field. From this, we show that the cycle classes can be represented by cycles that possess certain finiteness, surjectivity, and smoothness properties. It plays a key role in showing that the crystalline cohomology of smooth varieties can be expressed in terms of algebraic cycles.