Alexander Petrov: On de Rham Cohomology in Positive Characteristic
Explore the intricacies of de Rham cohomology in positive characteristic through this in-depth lecture by Alexander Petrov. Delve into the analog of Hodge decomposition established by Deligne and Illusie for smooth proper varieties over F_p with lifts over Z/p^2. Examine the conditions under which natural isomorphisms between de Rham and Hodge cohomology exist and investigate cases where these isomorphisms may fail for liftable varieties of higher dimensions. Analyze the role of non-vanishing cohomology of reductive groups in positive characteristics and the differing behaviors of Steenrod operations on de Rham and Hodge cohomology. Discover additional structures present on the de Rham complex of varieties over F_p, including the Sen operator in the context of lifts over Z/p^2 and the canonical decomposition after Frobenius pullback. Gain valuable insights into this complex area of algebraic geometry and cohomology theory.