English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this article, we suggest a categorification procedure in order to capturean analogy between Crystalline Grothendieck-Lefschetz trace formula and thecyclotomic trace map $K\rightarrow TC$ from the algebraic $K$-theory to thetopological cyclic homology $TC$. First, we categorify the category of schemesto the $(2, \infty)$-category of noncommuatative schemes a la Kontsevich. Thisgives a categorification of the set of rational points of a scheme. Then, wecategorify the Crystalline Grothendieck-Lefschetz trace formula and find ananalogue to the Crystalline cohomology in the setting of noncommuative schemesover $\mathbf{F}_{p}$. Our analogy suggests the existence of a categorificationof the $l$-adic cohomology trace formula in the noncommutative setting for$l\neq p$. Finally, we write down the corresponding dictionary.

Analogy between the cyclotomic trace map $K \rightarrow TC$ and the Grothendieck trace formula via noncommutative geometry