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Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$.

Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This gives an induced map $H^*(SU(n),k) \to H^*(G,k)$, with the following properties:

  • $H^*(SU(n),k)$ is a regular ring, and
  • $H^*(G,k)$ is finitely-generated as a module over $H^*(SU(n),k)$.

Now suppose that $G$ is a finite group scheme. Let $kG$ be the group algebra associated to $G$ (i.e., the $k$-linear dual of the coordinate algebra). The category of $kG$-modules is abelian with enough injectives, and for a $kG$-module $M$ we write $H^*(G,M) = \mathop{Ext}_{kG}(k,M)$. It is known by Friedlander-Suslin that $H^*(G,k)$ is a finitely-generated $k$-algebra, and $H^*(G,M)$ is a finite $H^*(G,k)$-module. I am after an analogue of the above result for finite group schemes. That is:

Given a finite group scheme $G$, does there exist a finite group scheme $H$ and a morphism $G \to H$ such that $H^*(H,k)$ is a regular ring and $H^*(G,k)$ is finitely-generated as a $H^*(H,k)$-module?

For any finite group scheme there is an embedding $G \to GL_n$ for some $n$. For an infinitesimal finite group scheme of height $r$, there is even an embedding into a Frobenius kernel $(GL_{n})_{(r)}$ for some $n$, however I could not find any result on the cohomology of these group schemes.

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