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 Cha\l upnik, Krause, and Touz\'e. 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 $\Ext$ 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 $\Ext$ groups for partitions with $2$ rows (or $2$ columns), and as such it relates to Parker's recursive description of $\Ext$ groups for $\SL_2$-representations. Finally, we give a general sharp bound for the interval of degrees where stable cohomology of a Schur functor can be non-zero.