Pro unitality and pro excision in algebraic K-theory and cyclic homology

TL;DR

This paper establishes pro excision properties in algebraic K-theory and cyclic homology, proving ideals of Noetherian rings are pro unital and extending key theorems in cyclic homology.

Contribution

It proves pro unitality of ideals in Noetherian rings and extends fundamental theorems in cyclic homology and K-theory using pro Tor vanishing conditions.

Findings

All ideals of Noetherian rings are pro unital.

Pro excision holds in K-theory, cyclic, and topological cyclic homology.

Extended theorems include a strong pro Hochschild--Kostant--Rosenberg and a generalised Fe
2fgin--Tsygan theorem.

Abstract

We study pro excision in algebraic K-theory, following Suslin--Wodzicki, Cuntz--Quillen, Corti\~nas, and Geisser--Hesselholt, as well as Artin--Rees and continuity properties of Andr\'e--Quillen, Hochschild, and cyclic homology. Our key tool is to first establish the equivalence of various pro Tor vanishing conditions which appear in the literature. Using this we prove that all ideals of commutative, Noetherian rings are pro unital in a certain sense, and show that such ideals satisfy pro excision in $K$-theory as well as in cyclic and topological cyclic homology. In addition, our techniques yield a strong form of the pro Hochschild--Kostant--Rosenberg theorem, an extension to general base rings of the Cuntz--Quillen excision theorem in periodic cyclic homology, and a generalisation of the Fe\u{\i}gin--Tsygan theorem.