Additive cycle complex and coherent duality

TL;DR

This paper constructs a cycle map linking additive cycle complexes to residual complexes in duality theory, leading to new injectivity results for higher Chow groups and motivic cohomology in positive characteristic.

Contribution

It introduces a new cycle map compatible with existing constructions, enhancing understanding of additive cycles and duality in algebraic geometry over fields of positive characteristic.

Findings

Constructed a cycle map from additive cycle complex to residual complex.

Proved injectivity for higher Chow groups with mod p coefficients.

Extended results to motivic cohomology with modulus over algebraically closed fields.

Abstract

Let $k$ be a field of positive characteristic $p$, and $X$ be a separated of finite type $k$-scheme of dimension $d$. We construct a cycle map from the additive cycle complex to the residual complex of Serre-Grothendieck coherent duality theory. This map is compatible with a cubical version of the map constructed in [Ren23] arXiv:2104.09662 when $k$ is perfect. As a corollary, we get injectivity statements for (additive) higher Chow groups as well as for motivic cohomology (with modulus) with $\mathbb{Z}/p$ coefficients when $k$ is algebraically closed.