Finite generation of cohomology for Drinfeld doubles of finite group schemes

We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any scheme, and D(G) ring kG, we show self-extension algebra trivial representation a algebra, each D(G)-representation V extensions from form module over aforementioned algebra. As corollary, find all categories \({{\,\mathrm{rep}\,}}(G)^*_\mathscr {M}\) dual \({{\,\mathrm{rep}\,}}(G)\) are also type (i.e. have cohomology), provide uniform bound on their Krull dimensions. This paper completes earlier work Friedlander author.