We study the coherent cohomology of generalized flag supervarieties. Our main observation is that these groups are closely related to the free resolutions of (certain generalizations of) determinantal ideals. In the case of super Grassmannians, we co
mpletely compute the cohomology of the structure sheaf: it is composed of the singular cohomology of a Grassmannian and the syzygies of a determinantal variety. The majority of the work involves studying the geometry of an analog of the Grothendieck-Springer resolution associated to the super Grassmannian; this takes place in the world of ordinary (non-super) algebraic geometry. Our work gives a conceptual explanation of the result of Pragacz-Weyman that the syzygies of determinantal ideals admit an action of the general linear supergroup. In a subsequent paper, we will treat other flag supervarieties in detail.
For a finite ring $R$, not necessarily commutative, we prove that the category of $text{VIC}(R)$-modules over a left Noetherian ring $mathbf{k}$ is locally Noetherian, generalizing a theorem of the authors that dealt with commutative $R$. As an appli
cation, we prove a very general twisted homology stability for $text{GL}_n(R)$ with $R$ a finite noncommutative ring.
