2015/03/01 by Ilias Amrani, Amrani, Ilias
Mathematics · #11M38 #14A22 #14F30 #19E08 #55N15 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #K-Theory and Homology (math.KT) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1503.00317
openalex publication_date 2015/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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→ TC from the algebraic K-theory to the topological cyclic homology TC. First, we categorify the category of schemes to the (2, ∞)-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 Fp. Our analogy suggests the existence of a categorification of the l-adic cohomology trace formula in the noncommutative setting for l≠ p. Finally, we write down the corresponding dictionary.