### On the p-typical curves in Quillen's K-theory, by Lars Hesselholt

Twenty 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, 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]/(t^2) of dual numbers over k and show that in
contrast to crystalline cohomology, its cohomology groups are finitely
generated modules over the Witt ring W(k).

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Lars Hesselholt < larsh@math.mit.edu>