Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.