Analytic cyclic homology in positive characteristic

TL;DR

This paper introduces a new cyclic homology theory for algebras over a field of positive characteristic, constructed via liftings to discrete valuation rings and suitable completions, with desirable invariance and stability properties.

Contribution

It defines a novel cyclic homology framework for positive characteristic fields using liftings to valuation rings and tube algebras, demonstrating key invariance and stability features.

Findings

The theory can be computed using any pro-dagger algebra lifting.

It is polynomially homotopy invariant.

It satisfies excision and matricial stability.

Abstract

Let $V$ be a complete discrete valuation ring with residue field $\mathbb{F}$. We define a cyclic homology theory for algebras over $\mathbb{F}$, by lifting them to free algebras over $V$, which we enlarge to tube algebras and complete suitably. We show that this theory may be computed using any pro-dagger algebra lifting of an $\mathbb{F}$-algebra. We show that our theory is polynomially homotopy invariant, excisive, and matricially stable.