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arXiv · 1710.06409

The geometry of the cyclotomic trace

David Ayala·Aaron Mazel-Gee·Nick Rozenblyum
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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.

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David Ayala, Aaron Mazel-Gee, Nick Rozenblyum. 2017-10-17. The geometry of the cyclotomic trace. https://arxiv.org/abs/1710.06409

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