# Has anyone used this theorem of P. Cartier?

In "Groupes Algebriques et Groupes Formels", Conf. au coll. sur la theorie des groupes algebriques, Bruxelles 1962, P. Cartier proves the following in Section 9, Theoreme 1:

(What follows is my English exposition/summary and translation)

Fix a field $k$ of characteristic $p$. Let $\phi: k \rightarrow k$ be the Frobenius endomorphism $\phi(x) = x^p$.

Let $U$ be the abelian category of unipotent commutative algebraic groups over $k$. For such a group $G$, let $G^{\phi} = G \times_{Spec(k)} Spec(k)$ be the fibre product, where $k$ is viewed as a $k$-algebra via $\phi$. There are Frobenius and Verschiebung homomorphisms for all $n > 0$: $$F^n: G \rightarrow G^{\phi^n}, V^n: G^{\phi^n} \rightarrow G.$$

Let $W_n$ be the group scheme of Witt vectors under addition, truncated so that the underlying scheme is just affine $n$-space. For any object $G$ of $U$, let $$H_n(G) = Hom(G, W_n).$$ (Cartier calls this functor $V_n$, but this is easily confused with Verschiebung!)

Let $U_n$ be the full subcategory of $U$ consisting of groups $G$ for which $V^n = 0$; the Witt group scheme $W_n$ is a prototypical example.

Let $E_n = Hom(W_n, W_n)$. This is a familiar Dieudonne ring (truncated). Then $H_n(G)$ is a contravariant functor from the category $U$ to the category of left $E_n$-modules.

Theorem 1, part (c): The functor $H_n$ is a duality from the category $U_n$ to the category of left $E_n$-modules.

Now, as Cartier notes afer the theorem, the category of left $E_n$-modules is well understood in the finite-length setting. This is the productive industry of Dieudonne modules. My question is...

Has anyone been using this theorem outside the finite-length setting? Certainly there are many naturally occurring commutative unipotent groups which are not finite over $k$. Does anyone use the corresponding left $E_n$-modules? If so, references would be appreciated? If not, why not -- are the left (say, finitely-generated) $E_n$-modules so complicated to study outside of the finite-length setting?

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