Nilpotent invariance of semi-topological K-theory of dg-algebras and the lattice conjecture

TL;DR

This paper proves the nilpotent invariance of semi-topological K-theory for certain dg-categories, establishing a natural rational structure on periodic cyclic homology and advancing the lattice conjecture.

Contribution

It demonstrates derived nilpotent invariance of semi-topological K-theory for proper connective dg-algebras and local systems, supporting the lattice conjecture.

Findings

Established nilpotent invariance of semi-topological K-theory

Constructed a natural rational structure on periodic cyclic homology

Validated the conjecture for specific classes of dg-categories

Abstract

We show existence of a natural rational structure on periodic cyclic homology, conjectured by L. Katzarkov, M. Kontsevich, T. Pantev, for several classes of dg-categories, including proper connective $\mathbb{C}$-dg-algebras and dg-categories of local systems. The main ingredient is derived nilpotent invariance of A. Blanc's semi-topological K-theory, which we establish along the way.