Cyclotomic trace

Remarks

The lift through $TC(R)$ exists because the Bökstedt trace lands in the part of $THH(R)$ that is fixed by the circle action and compatible with the Frobenius maps. Blumberg, Gepner and Tabuada showed that the cyclotomic trace is the unique multiplicative natural transformation from $K$ to $TC$, up to homotopy. Composing the Bökstedt trace with the linearization $THH(R) \to HH(R)$ gives on homotopy groups the Dennis trace map. Example: for $R = \mathbb{F}_p$, after $p$-completion, the cyclotomic trace on $\pi_0$ is an isomorphism from $K_0(\mathbb{F}_p) \otimes \Z_p \cong \Z_p$ to $\pi_0 TC(\mathbb{F}_p)^\wedge_p \cong \Z_p$. Non-example: the cyclotomic trace is not an equivalence in general; for $R = \mathbb{F}_p$ the group $\pi_{-1} TC(\mathbb{F}_p)^\wedge_p$ is not zero, while $K_{-1}(\mathbb{F}_p) = 0$.