The Topological Cartier--Raynaud Ring

We prove that the -category of p-typical topological Cartier modules, recently introduced by Antieau--Nikolaus, over some base A is equivalent to the -category of modules over a ring spectrum RA, which we call the topological Cartier--Raynaud ring. Our main result is an identification of the homotopy groups of RA. In particular, for A=W (k), the Witt vectors over k, the homotopy groups _* Rₖ (₊) 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.