An Alpine Bouquet of Algebraic Topology

Abstract

From a bimodule $M$ over an exact category $C$, we define an exact category $C\ltimes M$ with a projection down to $C$. This construction classifies certain split square zero extensions of exact categories. We show that the trace map induces an equivalence between the relative $K$-theory of $C\ltimes M$ and its relative topological cyclic homology. When applied to the bimodule $\hom(-,-\otimes_AM)$ on the category of finitely generated projective modules over a ring $A$ one recovers the classical Dundas-McCarthy theorem for split square zero extensions of rings.