Stable sheaf cohomology and Koszul--Ringel duality

We identify a close relationship between stable sheaf cohomology for polynomial functors applied to the cotangent bundle on projective space, and Koszul–Ringel duality on the category of strict polynomial functors as described in the work of Chałupnik, Krause, and Touzé. Combining this with recent results of Maliakas–Stergiopoulou we confirm a conjectured periodicity statement for stable cohomology. In a different direction, we find a remarkable invariance property for $\text{Ext}$ groups between Schur functors associated to hook partitions, and compute all such extension groups over a field of arbitrary characteristic. We show that this is further equivalent to the calculation of $\text{Ext}$ groups for partitions with $2$ rows (or $2$ columns), and as such it relates to Parker’s recursive description of $\text{Ext}$ groups for ${\text{SL}}_2$-representations. Finally, we give a general sharp bound for the interval of degrees where stable cohomology of a Schur functor can be non-zero.