Pro cdh-descent for cyclic homology and $K$-theory

TL;DR

This paper establishes that cyclic homology, topological cyclic homology, and algebraic K-theory satisfy a pro Mayer--Vietoris property for blow-up squares of varieties, supporting the well-definedness of K-theory with compact support.

Contribution

It proves a pro cdh-descent property for these theories in both zero and finite characteristic, extending the understanding of their behavior under blow-ups.

Findings

Pro Mayer--Vietoris property holds for cyclic homology, topological cyclic homology, and K-theory.

Supports the well-definedness of K-theory with compact support.

Applicable in both zero and finite characteristic settings.

Abstract

In this paper we prove that cyclic homology, topological cyclic homology, and algebraic $K$-theory satisfy a pro Mayer--Vietoris property with respect to abstract blow-up squares of varieties, in both zero and finite characteristic. This may be interpreted as the well-definedness of $K$-theory with compact support.