This course provides an overview of algebraic de Rham cohomology and crystalline cohomology, focusing on their applications in algebraic geometry. By the end of the course, learners will understand the computation of de Rham cohomology for smooth manifolds, the advantages and drawbacks of algebraic de Rham cohomology, and the construction and drawbacks of crystalline cohomology. The course teaches skills such as analyzing chain complexes, understanding saturated Dieudonné algebras, and sketching alternative constructions. The teaching method involves a lecture format with a focus on theoretical concepts and mathematical proofs. This course is intended for individuals interested in advanced topics in algebraic geometry and algebraic topology.
Overview
Syllabus
Intro
de Rham Cohomology for Smooth Manifolds
Example: The Variety C
Advantages of Algebraic de Rham Cohomology
The Algebraic de Rham Complex
Algebraic de Rham Cohomology in Positive Characteristic
Failure of Functoriality
Crystalline Cohomology
Drawbacks of the Crystalline Theory
Drawbacks of the de Rham-Witt Complex
Alternative Approach
Saturated Dieudonné Algebras
Proof Sketch
Conclusion
Non-Example: Completed de Rham Complexes
Taught by
Institute for Advanced Study
