In-Person Talk 

Let X/k be a smooth scheme over a perfect field k of characteristic p > 0 and let W be the Witt ring of k. Inspired by Drinfeld’s “stacky” approach to prismatic cohomology, Bhatt, Lurie, and others have shown that a lifting of X/k to W induces a canonical action of µ_p (and more) on the de Rham complex of X/k, in the derived category. The associated Z/pZ-grading generalizes the Deligne-Illusie decomposition theorem. I will explain an (only partially) successful attempt to find an explicit geometric approach to this construction and its categorification. This will not be a formal research talk, but rather a story about a struggle still in progress.