ABSTRACT
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 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math15_1.jpg"/> 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 ] https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/umath15_1.jpg"/> is a polynomial algebra on a generator x in degree two [10]. By comparison, HH ∗ ( F p / Z ) = F p 〈 x 〉 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/umath15_2.jpg"/> is the divided power algebra, 1 so Bökstedt’s periodicity theorem indeed shows that by replacing the base Z https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math15_2.jpg"/> by the base S https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math15_3.jpg"/> , denominators disappear. In fact, the base-change map HH ∗ ( F p / S ) → HH ∗ ( F p / Z ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math15_4.jpg"/> 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 ) , https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/umath15_3.jpg"/> so apparently the stable homotopy groups of spheres have exactly the right size to eliminate the denominators in the divided power algebra.
