The kernel of the Gysin homomorphism for positive characteristic

TL;DR

This paper extends a theorem on the Gysin kernel from complex numbers to uncountable algebraically closed fields of positive characteristic, using étale cohomology and comparison theorems.

Contribution

It generalizes the Gysin kernel theorem from complex surfaces to positive characteristic fields by adapting existing proofs and employing étale cohomology techniques.

Findings

The Gysin kernel theorem holds over uncountable algebraically closed fields of positive characteristic.

Key results are preserved using étale base change and comparison theorems.

The approach confirms the theorem's validity beyond complex algebraic geometry.

Abstract

Let $k$ be an uncountable algebraically closed field of positive characteristic and let $S$ be a smooth projective connected surface over $k$. We extend the theorem on the Gysin kernel from [20, Theorem 5.1] to also be true over $k$, where it was proved over $\mathbb{C}$. This is done by showing that almost all results still hold true over $k$ via the same argument or by using \'{e}tale base arguments and then using a lift with the Comparison theorems [16, Theorems 21.1 & 20.5] and Tate's Conjecture for finitely generated fields [27] and [31] as needed.