Topological cyclic homology

Authored by: Lars Hesselholt , Thomas Nikolaus

Handbook of Homotopy Theory

Print publication date:  December  2019
Online publication date:  December  2019

Print ISBN: 9780815369707
eBook ISBN: 9781351251624
Adobe ISBN:


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Topological cyclic homology is a manifestation of Waldhausen’s vision that the cyclic theory of Connes and Tsygan should be developed with the initial ring S of higher algebra as base. In his philosophy, such a theory should be meaningful integrally as opposed to rationally. Bökstedt realized this vision for Hochschild homology [9], and he made the fundamental calculation that THH ∗ ( F p ) = HH ∗ ( F p / S ) = F p [ x ] is a polynomial algebra on a generator x in degree two [10]. By comparison, HH ∗ ( F p / Z ) = F p ⟨ x ⟩ is the divided power algebra, 1 so Bökstedt’s periodicity theorem indeed shows that by replacing the base Z by the base S , denominators disappear. In fact, the base-change map HH ∗ ( F p / S ) → HH ∗ ( F p / Z ) can be identified with the edge homomorphism of a spectral sequence E i , j 2 = HH i ( F p / π ∗ ( S ) ) j ⇒ HH i + j ( F p / S ) , so apparently the stable homotopy groups of spheres have exactly the right size to eliminate the denominators in the divided power algebra.

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