A Gentle Approach to Crystalline Cohomology

Let X be a smooth affine algebraic variety over the field C of complex numbers (that is, a smooth submanifold of C^n which can be described as the solutions to a system of polynomial equations). Grothendieck showed that the de Rham cohomology of X can be computed using only polynomial differential forms on X. This observation was the starting point for the theory of algebraic de Rham cohomology, which has proved to be a useful invariant for algebraic varieties over an arbitrary field k. In the case where k has positive characteristic, Berthelot and Grothendieck introduced a refinement of algebraic de Rham cohomology, known as crystalline cohomology. Later work of Bloch, Deligne, and Illusie showed that crystalline cohomology could be computed using an explicit chain complex, called the de Rham-Witt complex. In this talk, I'll give an overview of some of these ideas and sketch an alternative construction of the de Rham-Witt complex (joint work with Bhargav Bhatt and Akhil Mathew).