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>