# Motives Seminar-SS-2018

## Algebraic K-theory and topological cyclic homology

New time and room: Tues., 16-18, WSC-N-U-3.04 (mostly, sometimes in the Tea Room

The goal this semester will be the main results of the recent paper of Clausen-Matthew-Morrow on $p$-adic $K$-theory and topological cyclic homology in characteristic $p$. Along the way we will go over a number of classical results on the relationship of algebraic $K$-theory and (topological) cyclic homology (Dundas-Goodwillie, McCarthy) and computations of mod $p$ $K$-theory in characteristic $p$ (Bloch-Kato, Levine-Geisser, Geisser-Hesselholt). We will also revisit some of these results from a more modern point of view (Bhatt-Morrow-Scholze).

## Program

Lecture 1. (17.04-Marc Levine). Mod $p$ Bloch-Kato and Beilinson-Lichtenbaum in characteristic $p$.
Lecture 2. (24.04-Fangzhou Jin) The theorem of Dundas-Goodwillie
Lecture 3. (08.05-Tom Bachman) McCarthy's refinement of Dundas-Goodwillie
Lecture 4. (22.05-Gabriela Guzman) [BMS2]: Foundations of $HH$, $THH$, $TC^-$, $TP$ and faithfully flat descent.
Lecture 5. (29.05-Lorenzo Mantovani). [BMS2]: computations of $THH$, $TC^-$, $TP$ of smooth algebras over a perfect ring of characteristic $p$.
Lecture 6. (05.06-Maria Yakeson). $TC_n$ is given by $W\Omega_{log}$, reproving Geisser-Hesselholt.
Lecture 7. (12.06-Alejo Lopez-Avila). [CMM, sections 2.1, 2.2, 2.3 ]The finiteness property of $TC$
Lecture 8. (19.06-Ran Azouri). [CMM section 3] Hensel pairs and Gabber rigidity
Lecture 9. (26.06) conference in Berlin
Lecture 10. (03.07-Enzo Serandon) [CMM sections 4.1, 4.2] Main rigidity results. Part I
Lecture 11. (10.07-Rakesh Pawar) [CMM sections 4.3, 4.4] Main rigidity results. Part II
Lecture 12. (17.07) [CMM section 5.1, 5.2] Applications.

For details as to the program of lectures, please see the program

## Bibliography

[BMS2] Bhargav Bhatt, Matthew Morrow, Peter Scholze, Topological Hochschild homology and integral $p$-adic Hodge theory arxiv:1802.0326.

[BK] Bloch, Spencer; Kato, Kazuya. $p$-adic \'etale cohomology. Inst. Hautes Études Sci. Publ. Math. No. 63 (1986), 107--152.

[CMM] D. Clausen, A. Matthew, M. Morrow. $K$-Theory and topological cyclic homology of henselian pairs. arxiv:1803.10897

[Gabber] O. Gabber. $K$-theory of Henselian local rings and Henselian pairs. In Algebraic $K$-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), volume 126 of Contemp. Math., pages 59--70. Amer. Math. Soc., Providence, RI, 1992.

[GH] T. Geisser and L. Hesselholt. On the $K$-theory of complete regular local $\mathbb{F}_p$-algebras. Topology, 45(3): 475--493, 2006.

[GL] T. Geisser and M. Levine.The $K$-theory of fields in characteristic $p$. Invent. Math., 139(3):459--493, 2000.

[Goodwillie] Goodwillie, Thomas G. Relative algebraic K-theory and cyclic homology. Ann. of Math. (2) 124 (1986), no. 2, 347–402.

[Kato] Kato, Kazuya. Galois cohomology of complete discrete valuation fields. Algebraic K-theory, Part II (Oberwolfach, 1980), pp. 215–238, Lecture Notes in Math., 967, Springer, Berlin-New York, 1982.

[McCarthy] R. McCarthy. {Relative algebraic K-theory and topological cyclic homology. Acta Math., 179(2):197–222, 1997.