Suslin's singular homology and cohomology

TL;DR

This paper explores Suslin's singular homology and cohomology, analyzing their properties over different fields, especially finite fields, and proposes a modified definition with desirable homological features.

Contribution

It introduces a modified Suslin homology theory that aligns with classical properties and extends to singular schemes, with focus on finite generation and behavior over various fields.

Findings

Finite generation properties are established.

Modified homology behaves like a classical homology theory.

Degree one relates to the abelianized tame fundamental group.

Abstract

We discuss Suslin's singular homology and cohomology. In the first half we examine the p-part in characteristic p, and the situation over non-algebraically closed fields. In the second half we focus on finite base fields. We study finite generation properties, and give a modified definition which behaves like a homology theory: in degree zero it is a copy of Z for each connected component, in degree one it is related to the abelianized (tame) fundamental group, even for singular schemes, and it is expected to be finitely generated in general.