Smooth functions on algebraic K-theory

TL;DR

This paper explores the relationship between smooth functions on algebraic K-theory presheaves and real Deligne cohomology, revealing dualities with homotopy fibers of the Chern character in complex algebraic and non-commutative geometry.

Contribution

It establishes a connection between smooth functions on K-theory presheaves and Deligne cohomology, and identifies their duality with the homotopy fiber of the Chern character.

Findings

Smooth functions on K-theory relate to real Deligne cohomology.

Duality with the homotopy fiber of the Chern character.

Applicable to complex schemes and non-commutative derived schemes.

Abstract

For any complex scheme X or any dg category, there is an associated K-theory presheaf on the category of complex affine schemes. We study real smooth functions on this presheaf, defined by Kan extension, and show that they are closely related to real Deligne cohomology. When X is quasi-compact and semi-separated, and for various non-commutative derived schemes, these smooth functions on K-theory are dual to the homotopy fibre of the Chern character from Blanc's semi-topological K-theory to cyclic homology.