On the cyclic homology of ringed spaces and schemes

TL;DR

This paper establishes a connection between the cyclic homology of schemes with ample line bundles and their vector bundle categories, providing a new approach to the Chern character of perfect complexes.

Contribution

It introduces a novel proof linking cyclic homology of schemes to their vector bundle categories and offers a new construction of the Chern character for perfect complexes.

Findings

Cyclic homology of schemes with ample line bundles matches that of their vector bundle categories.

New construction method for the Chern character of perfect complexes.

Enhanced understanding of cyclic homology in algebraic geometry.

Abstract

We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles. As a byproduct of the proof, we obtain a new construction of the Chern character of a perfect complex on a ringed space.