Title of the mini-course: The Frobenius twist in functor cohomology by Vincent Franjou(Université de Nantes).
Abstract
Frobenius twist changes a representation in positive characteristic p by base change through the Frobenius map (or p-th power map). For representations of reductive algebraic groups, such as the General Linear Group, the resulting map in cohomology is injective by a result of Andersen. Cline, Parshall, Scott and van der Kallen [1977] showed that the algebraic group cohomology of iterated Frobenius twist eventually compares with the discrete group cohomology of the associated Chevalley group.
At the end of the last century, functor techniques allowed to give computational precision to such statements. The precise understanding in cohomology of the Frobenius twist for strict polynomial functors recently lead Touzé to a second proof of the van der Kallen conjecture, that reductive groups have finitely generated cohomology algebras.
This course will start with a short presentation of strict polynomial functors and Schur algebras. Classical cohomological computations for strict polynomial functors will follow. Touzé’s explicit vision of the Frobenius twist in cohomology will be presented. This leads to the statement of Touzé’s formality conjecture, recently proved by Chalupnik. For smoother statements, a short presention of derived categories will be included. This should allow to present van der Kallen and Touzé’s own argument for proving the formality conjecture. This conjecture (or even a weaker form due to Touzé) provides the new proof of finite cohomological generation for reductive groups.
Tentative dates
Friday, August 16, 2013
4:00 pm – 5:30 pm
Monday, August 19, 2013
4:00 pm – 5:30 pm
Wednesday, August 21, 2013
4:00 pm – 5:30 pm
Monday, August 26, 2013
4:00 pm – 5:30 pm
Venue: Vietnam Institute for Advanced Study in Mathematics.
- The schedule will be adjusted according to the possibilities of the participants
- All interested are invited to participants in the mini-course