Non-Archimedean analytic cyclic homology

TL;DR

This paper introduces a new form of analytic cyclic homology for certain algebras over a valuation ring, establishing key properties and linking it to rigid cohomology in specific cases.

Contribution

It defines analytic cyclic homology for complete torsion-free bornological algebras over a valuation ring and proves its fundamental invariance properties.

Findings

Homotopy invariance established

Computed for tensor products with Leavitt path algebras

Identified with Berthelot's rigid cohomology for smooth curves

Abstract

Let $V$ be a complete discrete valuation ring with fraction field $F$ of characteristic zero and with residue field $\mathbb{F}$. We introduce analytic cyclic homology of complete torsion-free bornological algebras over $V$. We prove that it is homotopy invariant, stable, invariant under certain nilpotent extensions, and satisfies excision. We use these properties to compute it for tensor products with dagger completions of Leavitt path algebras. If $R$ is a smooth commutative $V$-algebra of relative dimension $1$, then we identify its analytic cyclic homology with Berthelot's rigid cohomology of $R\otimes_{V}\mathbb{F}$.