Coalgebraic $K$-theory
Teena Gerhardt, Maximilien Péroux, W. Hermann B. Soré
TL;DR
The work develops a coalgebraic analogue K^c(C) of algebraic K-theory for coalgebras C over a field, defined via finitely cogenerated injectives, and its G-theory counterpart G^c(C) via finite-dimensional comodules. It constructs natural comparison maps K^c(C) -> K(C^*) and G^c(C) -> G^k(C^*) and proves they are equivalences under finite-dimensionality or Noetherian hypotheses, with K^c(C) inheriting ring structures when C is cocommutative. The paper also analyzes the Sweedler dual A^° to relate K^c(A^°) with K(A) (and G^c(A^°) with G^k(A)), giving precise conditions for when these maps are inverses, including key instances such as K^c(k⟨X⟩) ≃ K(k[[y]]). It culminates in applications to Swan theory, showing G^c(R_k(Γ)) ≃ G^k(kΓ) and expressing group characters within a coalgebraic trace framework via coHochschild theory. Overall, the results provide duality-driven bridges between algebraic K/G-theory and their coalgebraic analogues, with concrete consequences for representations and group theory.
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.