Algebraic $K$-theory, cohomotopy $K$-groups, and Koszul duality

TL;DR

This paper explores the relationship between algebraic K-theory, cohomotopy K-groups, and Koszul duality for augmented differential graded algebras, proposing a concrete candidate for Loday's conjectural K-groups.

Contribution

It combines derived Koszul duality with the Jones-Goodwillie Chern character to connect K-theory of certain categories with Loday's conjectural K-groups.

Findings

Established an equivalence between K-theory of thick subcategories and derived categories of Koszul duals.

Connected K-groups to Loday's conjectural contravariant K-groups using classical isomorphisms.

Proposed a concrete candidate for Loday's K-groups based on this framework.

Abstract

Let $A$ be an augmented differential graded algebra over a field $k$ of characteristic zero, and let $A^!=\mathbf{R}\mathrm{Hom}_A(k,k)$ be its Koszul dual algebra. Blumberg and Mandell showed that, under some finiteness conditions of $A$, the derived Koszul duality provides an equivalence between the $K$-theory $K(\mathrm{thick}_A(k))$ of the triangulated thick subcategory generated by $k$ and the $K$-theory $K(A^!)$ of the derived category of perfect $A^!$-modules. Combining this equivalence with the Jones-Goodwillie Chern character and the Jones-McCleary isomorphism, we obtain that the $K$-groups $K_n(\mathrm{thick}_A(k))$ are a concrete candidate for Loday's conjectural contravariant $K$-groups.