Topological cyclic homology

Definition

Let $R$ be a connective associative ring spectrum, and let $T$ be the circle group. Let $X = THH(R)$ be its topological Hochschild homology, with its $T$-action. For a group $G$ acting on a spectrum $Y$, write $Y^{hG}$ for the homotopy fixed points. For finite $G$, write $Y^{tG}$ for the Tate construction, the cofiber of the norm map $Y_{hG} \to Y^{hG}$. The spectrum $X$ is cyclotomic: for each prime $p$ there is a $T$-equivariant Frobenius map $\varphi_p \colon X \to X^{tC_p}$, where $C_p \subseteq T$ is the subgroup of order $p$. Applying $(-)^{hT}$ gives maps $\varphi_p^{hT} \colon X^{hT} \to (X^{tC_p})^{hT}$. There are also canonical maps $\mathrm{can}_p \colon X^{hT} \to (X^{tC_p})^{hT}$. The topological cyclic homology of $R$ is the fiber $$TC(R) = \operatorname{fib}\Big( X^{hT} \xrightarrow{\ (\varphi_p^{hT} - \mathrm{can}_p)_p\ } \prod_p (X^{tC_p})^{hT} \Big).$$