The Topological Cartier--Raynaud Ring

We prove that the $\mathrm{\infty }$-category of $p$-typical topological Cartier modules, recently introduced by Antieau–Nikolaus, over some base $A$ is equivalent to the $\mathrm{\infty }$-category of modules over a ring spectrum ${\mathcal{R}}_A$, which we call the topological Cartier–Raynaud ring. Our main result is an identification of the homotopy groups of ${\mathcal{R}}_A$. In particular, for $A=W(k)$, the Witt vectors over $k$, the homotopy groups ${\pi }_{\ast }{\mathcal{R}}_{W(k)}$ recover the classical Cartier–Raynaud ring constructed by Illusie–Raynaud. Moreover, along the way we will describe the compact generator of $p$-typical topological Cartier modules and classifies all natural operations on homotopy groups of $p$-typical topological Cartier modules.