Explore a gentle approach to crystalline cohomology in this 56-minute lecture by Jacob Lurie, Professor at the School of Mathematics, Institute for Advanced Study. Delve into the foundations of de Rham cohomology for smooth affine algebraic varieties, tracing its evolution from Grothendieck's observations to the development of algebraic de Rham cohomology. Examine the refinement of this theory in positive characteristic fields through crystalline cohomology, introduced by Berthelot and Grothendieck. Discover the significance of the de Rham-Witt complex in computing crystalline cohomology, as revealed by Bloch, Deligne, and Illusie. Follow Lurie's overview of these concepts and his presentation of an alternative construction of the de Rham-Witt complex, based on joint work with Bhargav Bhatt and Akhil Mathew. Gain insights into topics such as saturated Dieudonné algebras and the limitations of completed de Rham complexes in this comprehensive exploration of advanced mathematical concepts.
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
