On the p-typical curves in Quillen's K-theory

Several years ago, Bloch introduced the complex C*(A;p) of p-typical curves in K-theory and outlined its connection to the crystalline cohomology of Berthelot-Grothendieck. However, to verify this connection Bloch restricted his attention to the symbolic part of K-theory, since only this admitted a detailed study at the time. In this paper we evaluate C*(A;p) in terms of Bokstedt's topological Hochschild homology. Using this we show that if A is a smooth algebra over a perfect field k of positive characteristic p, then C*(A;p) is isomorphic to the de Rham-Witt complex of Bloch-Deligne-Illusie. This confirms the outlined relationship between p-typical curves in K-theory and crystalline cohomology in the smooth case. In the singular case, however, we get something new: we calculate C*(A;p) for the ring k[t]/(t2) of dual numbers over k and show that, by contrast to crystalline cohomology, its cohomology groups are finitely generated modules over the ring of p-typical Witt vectors W(k).

Lars Hesselholt <larsh@math.nagoya-u.ac.jp>