From Betti cohomology to crystalline cohomology

Abstract: I will first give a brief overview of relations between Betti, de Rham, and ℓ-adic cohomologies in local arithmetic situations.

I will then describe the origin, definition, and main properties of crystalline cohomology, and motivate the construction of de Rham-Witt complexes.

This colloquium lecture is aimed at a general audience.