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

Abstract

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