We consider a finite group scheme G, and its associated representation category rep G.  Here one can think of a finite discrete group, or an infinitesimal group scheme, such as the kernel of the r-th Frobenius map for GL_n over F_p.  Via a standard tensor categorical construction one has Drinfeld’s center Z(rep G) of the category of G-representations, which provides the universal central tensor functor Z(rep G) -> rep G.  I will describe recent work in which we prove that cohomology for the center of rep G satisfies strong finiteness properties.  In particular, the self-extension algebra of the unit 1 in Z(rep G) forms a finitely generated algebra over the base field, and for any other object V, extensions from 1 to V form a finitely generated module over this algebra.  I will explain some of the mathematics behind this result, and also provide the relevant historical framing.  Parts of this project are joint with Eric Friedlander.