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

TL;DR

This paper proposes a categorification framework linking the Grothendieck-Lefschetz trace formula with the cyclotomic trace map in algebraic K-theory, using noncommutative geometry to draw deep analogies.

Contribution

It introduces a categorification of schemes and trace formulas, establishing a novel analogy between crystalline cohomology and noncommutative geometry in algebraic K-theory.

Findings

Categorification of schemes to noncommutative schemes.

Analogy between crystalline cohomology and noncommutative geometry.

Potential for categorifying the l-adic cohomology trace formula.

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.