Topological cyclic homology

TY - CHAP

T1 - Topological cyclic homology

AU - Hesselholt, Lars

AU - Nikolaus, Thomas

PY - 2019/12/18

Y1 - 2019/12/18

N2 - This handbook chapter gives an introduction to topological cyclic homology following the modern setup of Nikolaus-Scholze. It contains a proof of the Bökstedt periodicity theorem close to his original unpublished proof, as well as the proof by Bhatt-Morrow-Scholze of its extension to all perfectoid rings. The latter is used to give a purely p-adic proof of the Bott periodicity theorem. Finally, the cofiber of the assembly map in p-adic K-theory for a cyclic group of order p and a perfectoid ring of coefficients is calculated.

AB - This handbook chapter gives an introduction to topological cyclic homology following the modern setup of Nikolaus-Scholze. It contains a proof of the Bökstedt periodicity theorem close to his original unpublished proof, as well as the proof by Bhatt-Morrow-Scholze of its extension to all perfectoid rings. The latter is used to give a purely p-adic proof of the Bott periodicity theorem. Finally, the cofiber of the assembly map in p-adic K-theory for a cyclic group of order p and a perfectoid ring of coefficients is calculated.

M3 - Book chapter

SN - 9780815369707

T3 - CRC Press/Chapman and Hall Handbooks in Mathematics Series

BT - Handbook of Homotopy Theory

A2 - Miller, Haynes

PB - CRC Press

ER -