The geometry of the cyclotomic trace

Abstract

We provide a new construction of the topological cyclic homology $TC(C)$ of any spectrally-enriched $\infty$-category $C$, which affords a precise algebro-geometric interpretation of the cyclotomic trace map $K(X)\to TC(X)$ 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.

References in corpus (4)

• Localization theorems in topological Hochschild homology and topological cyclic homology
• The smooth Whitehead spectrum of a point at odd regular primes
• The homotopy groups of the algebraic K-theory of the sphere spectrum
• Fibrations of $\infty$-categories

Cited by in corpus (5)

• K-theory and topological cyclic homology of henselian pairs
• Kaledin's degeneration theorem and topological Hochschild homology
• Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra
• A geometric approach to equivariant factorization homology and nonabelian Poincaré duality
• Spectral Waldhausen categories, the $S_{\bullet }$-construction, and the Dennis trace