The topological Cartier–Raynaud ring

We prove that the ∞ \infty -category of p p -typical topological Cartier modules, recently introduced by Antieau–Nikolaus, over some base A A is equivalent to the ∞ \infty -category of modules over a ring spectrum R A \mathcal R_A , which we call the topological Cartier–Raynaud ring. Our main result is an identification of the homotopy groups of R A \mathcal R_A . In particular, for A = W ( k ) A=W(k) the homotopy groups π ∗ R W ( k ) \pi _*\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 p -typical topological Cartier modules and classify all natural operations on homotopy groups of p p -typical topological Cartier modules.