Coalgebraic K-theory

2025/03/06 by Teena Gerhardt, Gerhardt, Teena, Maximilien Péroux +3 · 2 voices
Mathematics · #19D50. Secondary: 18M05 #55U30 #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Primary: 16T15 #math.AT #math.KT

paper · pdf · doi:10.48550/arxiv.2503.04897

arxiv published 2025/03/06 · arxiv updated 2026/04/21

Abstract

We establish comparison maps between the classical algebraic K-theory of algebras over a field and its analogue K^c, an algebraic K-theory for coalgebras over a field. The comparison maps are compatible with the Hattori--Stallings (co)traces. We identify conditions on the algebras or coalgebras under which the comparison maps are equivalences. Notably, the algebraic K-theory of the power series ring is equivalent to the K^c-theory of the divided power coalgebra. We also establish comparison maps between the G-theory of finite dimensional representations of an algebra and its analogue G^c for coalgebras. In particular, we show that the Swan theory of a group is equivalent to the G^c-theory of the representative functions coalgebra, reframing the classical character of a group as a trace in coHochschild homology.

Discussions

• the $G^c$-theory of the representative functions coalgebra, reframing the classical character of a group as a trace in coHochschild homology. [4/4 of https://arxiv.org/abs/2503.04897v1] [bsky, 1 points, 0 comments]
• Teena Gerhardt, Maximilien P\'eroux, W. Hermann B. Sor\'e Coalgebraic $K$-theory https://arxiv.org/abs/2503.04897 [bsky, 0 points, 0 comments]

Related

• Trace methods for coHochschild homology
• K-theory and derived equivalences
• Realizing coalgebras over the Steenrod algebra
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• Coalgebras and quantization