Joint IAS/PU Arithmetic Geometry

I will discuss new structural properties of the de Rham cohomology of smooth schemes over the ring of Witt vectors. The main technical input is the $F$-gauge structure on crystalline cohomology. First, I will explain the slope obstruction to the injectivity of the de Rham-to-crystalline comparison morphism. This yields a negative answer to a question posed by Esnault–Kisin–Petrov. On the positive side, I will show that the comparison morphism becomes injective when restricted to suitable subspaces defined by slope conditions. Finally, I will illustrate how these techniques allow us to determine the de Rham cohomology modulo torsion for such schemes. This is joint work with Daniel Caro.