On the relative and bi-relative K-theory of rings of finite characteristic

TL;DR

This paper investigates the structure of relative and bi-relative algebraic K-theory groups over rings of finite characteristic, showing they are p-primary torsion groups with bounded exponent, and extends the cyclotomic trace map to a broader K-theory context.

Contribution

It establishes the torsion properties of relative and bi-relative K-groups over Z/p^N Z-algebras and extends the cyclotomic trace map to Bass completed non-connective K-theory.

Findings

Relative and bi-relative K-groups are p-primary torsion with bounded exponent.

The cyclotomic trace map extends to Bass completed non-connective K-theory.

Provides structural insights into K-theory over rings of finite characteristic.

Abstract

We prove that the relative K-groups associated with a nilpotent extension of Z/p^N Z-algebras and the bi-relative K-groups associated with a Milnor square of Z/p^N Z-algebras are p-primary torsion groups of bounded exponent. We also show that, in general, the cyclotomic trace map extends from Quillen K-theory to Bass completed non-connective algebraic K-theory.