Dieudonné module

In mathematics, a Dieudonné module introduced by Jean Dieudonné (1954, 1957b), is a module over the non-commutative Dieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms F and V called the Frobenius and Verschiebung operators. They are used for studying finite flat commutative group schemes.If k is a field of characteristic p, its ring of Witt vectors consists of sequences (w1, w2, w3, ...) of elements of k, and has an endomorphism σ induced by the Frobenius endomorphism of k, so (w1, w2, w3, ...)σ = (wp1, wp2, wp3, ...). The Dieudonné ring, often denoted by Ek or Dk, is the non-commutative ring over W(k) generated by 2 elements F and V subject to the relationsSpecial sorts of modules over the Dieudonné ring correspond to certain algebraic group schemes. For example, finite length modules over the Dieudonné ring form an abelian category equivalent to the opposite of the category of finite commutative p-group schemes over k.The Dieudonné–Manin classification theorem was proved by Dieudonné (1955) and Yuri Manin (1963). It describes the structure of Dieudonné modules over an algebraically closed field k up to 'isogeny'. More precisely, it classifies the finitely generated modules over D k [ 1 / p ] {displaystyle D_{k}}  , where D k {displaystyle D_{k}}   is the Dieudonné ring. The category of such modules is semisimple, so every module is a direct sum of simple modules. The simple modules are the modules Es/r where r and s are coprime integers with r>0. The module Es/r has a basis over W(k) of the form v, Fv, F2v,...,Fr−1v for some element v, and Frv = psv. The rational number s/r is called the slope of the module.If G is a commutative group scheme, its Dieudonné module D(G) is defined to be Hom(G,W), defined as limn Hom(G, Wn) where W is the formal Witt group scheme and Wn is the truncated Witt group scheme of Witt vectors of length n. A Dieudonné crystal is a crystal D together with homomorphisms F:Dp→D and V :D→Dp satisfying the relations VF=p (on Dp), FV=p (on D). Dieudonné crystals were introduced by Grothendieck (1966). They play the same role for classifying algebraic groups over schemes that Dieudonné modules play for classifying algebraic groups over fields.