On a theorem of Keller over a base ring

TL;DR

This paper generalizes Keller's theorem, demonstrating that the cyclic homology of a quasi-compact separated scheme over a base ring is canonically isomorphic to that of its perfect complexes, highlighting the categorical nature of cyclic homology.

Contribution

The paper extends Keller's theorem from schemes over a field to schemes over a base commutative ring, broadening its applicability.

Findings

Cyclic homology of schemes over a base ring is isomorphic to that of their perfect complexes.

The categorical nature of cyclic homology is confirmed in a more general setting.

Generalization of Keller's theorem to schemes over arbitrary base rings.

Abstract

Let $X$ be a quasi-compact separated scheme over a base field. Keller proved a theorem stating that the cyclic homology of $X$ is canonically isomorphic to the cyclic homology of the dg category ${\sf Perf}(X)$ consisting of perfect complexes over $X$. This theorem shows the categorical nature of the cyclic homology. In this note, we generalize Keller's theorem to allow $X$ be defined over a base commutative ring.