Coalgebraic $K$-theory

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.