University of Sydney Algebra Seminar

William Hardesty (University of Sydney)

Friday 20 September, 12-1pm, Place: Carslaw 375

Co-t-structures on derived categories of coherent sheaves and the cohomology of tilting modules

Since the mid 1990s, understanding the “Frobenius kernel cohomology” of tilting modules has been an important area of research in modular representation theory. Another important area of research, which is sometimes called “coherent Springer theory”, involves the study of derived equivariant coherent sheaves on the nilpotent cone and the Springer resolution. Due to the work of Achar and Riche, we now know that both of these topics are related by a derived equivalence. In particular, the cohomology of a tilting module is given by the global sections of a corresponding (derived) equivariant coherent sheaf on nilpotent cone. In this talk, I will present recent joint work with Pramod Achar, where we intrinsically characterize these sheaves as the indecomposable objects in the “co-heart” of a certain non-trivial “co-t-structure” on the equivariant derived category of coherent sheaves on the nilpotent cone. I will also discuss some additional exciting progress that we have towards the study of tilting module cohomology by employing the framework of co-t-structures. This includes a new proof of the “Humphreys conjecture on support varieties” for \(SL_n\), as well as a conjecture which alternatively characterizes these sheaves as extensions of “tilting bundles” on nilpotent orbits.