Topological cyclic homology

By L. Hesselholt, T. Nikolaus. Handbook of Homotopy Theory, 2019

311 citations recorded by the source corpus.

This survey of topological cyclic homology is a chapter in the Handbook on Homotopy Theory. We give a brief introduction to topological cyclic homology and the cyclotomic trace map following Nikolaus-Scholze, followed by a proof of B\"okstedt periodicity that closely resembles B\"okstedt's original unpublished proof. We explain the extension of B\"{o}kstedt periodicity by Bhatt-Morrow-Scholze from perfect fields to perfectoid rings and use this to give a purely p-adic proof of Bott periodicity. Finally, we evaluate the cofiber of the assembly map in p-adic topological cyclic homology for the cyclic group of order p and a perfectoid ring of coefficients.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/53be251f8eaf26027db63a46052b1d8a7dd86684

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. B. Bhatt
2. M. Morrow
3. Peter Scholze
4. Dustin Clausen
5. A. Mathew
6. Lue Pan
7. Ruochuan Liu
8. Kiran S. Kedlaya

In Geometry and Topology

1. B. Bhatt
2. Peter Scholze
3. A. Mathew
4. Dustin Clausen
5. M. Morrow
6. Ruochuan Liu
7. Kiran S. Kedlaya
8. Wiesława Nizioł

In Mathematical Physics

1. B. Bhatt
2. Peter Scholze
3. A. Mathew
4. Dustin Clausen
5. M. Morrow
6. Ruochuan Liu
7. Kiran S. Kedlaya
8. Wiesława Nizioł

Citations in the corpus

• Prisms and prismatic cohomology (2019, background): “ quasiregular semiperfectoid ring (see Theorem 1.11). The formalism of topological Hochschild homology endows the commutative ring π0TP(S;Zp) with an endomorphism φS often called the “Frobenius” (see [NS18]). Despite the name, it is not clear from the definitions that φSlifts the Frobenius on π0TP(S;”
• K-theory and topological cyclic homology of henselian pairs (2018, methodology): “In Remark 2.14 below, we will also give a proof purely based on the Nikolaus–Scholze approach.”
• p-ADIC GEOMETRY (2017, background): “lation, THH has even more structure as encoded in the structure of a cyclotomic spectrum. However, in joint work with Nikolaus, we proved that this extra structure is actually redundant. Theorem 8.4 ([NS17]). On respective subcategories of bounded below objects, the 1-category of cyclotomic spectra ”