The geometry of the cyclotomic trace

2017/10/17 by Ayala, David, Mazel-Gee, Aaron, Rozenblyum, Nick
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.1710.06409

Abstract

We provide a new construction of the topological cyclic homology TC(C) of any spectrally-enriched ∞-category C, which affords a precise algebro-geometric interpretation of the cyclotomic trace map K(X) → TC(X) from algebraic K-theory to topological cyclic homology for any scheme X. This construction rests on a new identification of the cyclotomic structure on THH(C), which we find to be a consequence of (i) the geometry of 1-manifolds, and (ii) linearization (in the sense of Goodwillie calculus). Our construction of the cyclotomic trace likewise arises from the linearization of more primitive data.

Related

• The cyclotomic trace for symmetric ring spectra
• On the equivalence of two theories of real cyclotomic spectra
• On topological cyclic homology
• Cyclotomic complexes
• Analogy between the cyclotomic trace map K → TC and the Grothendieck trace formula via noncommutative geometry