Dieudonné module
Encyclopedia
In mathematics, a Dieudonné module introduced by , is a module
over the non-commutative Dieudonné ring, which is generated over the ring of Witt vector
s by two special endomorphisms F and V called the Frobenius and Verschiebung operators. They are used for studying finite flat commutative group schemes.
Finite flat commutative group schemes over a perfect field
k of positive characteristic p can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring D = W(k){F,V}/(FV − p), which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of k. F and V are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors. Jean Dieudonné
and Pierre Cartier
constructed an antiequivalence of categories between finite commutative group schemes over k of order a power of "p" and modules over D with finite W(k)-length. The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf CW of Witt co-vectors. This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive Verschiebung maps V: Wn → Wn+1, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected p-group schemes correspond to D-modules for which F is nilpotent, and étale group schemes correspond to modules for which F is an isomorphism.
Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze p-divisible groups. Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in Wiles
's work on the Shimura-Taniyama conjecture.
s 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 = (w,w,w,...). 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 relations
It is a Z-graded ring, where the piece of degree n∈Z is a 1-dimensional free module over W(k), spanned by V−n if n≤0 and by Fn if n≥0.
Some authors define the Dieudonné ring to be the completion of the ring above for the ideal generated by F and V.
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over the non-commutative Dieudonné ring, which is generated over the ring of Witt vector
Witt vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order p is the ring of p-adic integers.-Motivation:Any p-adic...
s by two special endomorphisms F and V called the Frobenius and Verschiebung operators. They are used for studying finite flat commutative group schemes.
Finite flat commutative group schemes over a perfect field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
k of positive characteristic p can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring D = W(k){F,V}/(FV − p), which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of k. F and V are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors. Jean Dieudonné
Jean Dieudonné
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
and Pierre Cartier
Pierre Cartier
Pierre Cartier may refer to:* Pierre Cartier * Pierre Cartier * Pierre Cartier...
constructed an antiequivalence of categories between finite commutative group schemes over k of order a power of "p" and modules over D with finite W(k)-length. The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf CW of Witt co-vectors. This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive Verschiebung maps V: Wn → Wn+1, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected p-group schemes correspond to D-modules for which F is nilpotent, and étale group schemes correspond to modules for which F is an isomorphism.
Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze p-divisible groups. Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in Wiles
Andrew Wiles
Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory...
's work on the Shimura-Taniyama conjecture.
Dieudonné rings
If k is a field of characteristic p, its ring of Witt vectorWitt vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order p is the ring of p-adic integers.-Motivation:Any p-adic...
s 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 = (w,w,w,...). 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 relations
- FV = VF = p
- Fw = wσF
- wV = Vwσ
It is a Z-graded ring, where the piece of degree n∈Z is a 1-dimensional free module over W(k), spanned by V−n if n≤0 and by Fn if n≥0.
Some authors define the Dieudonné ring to be the completion of the ring above for the ideal generated by F and V.
Dieudonné modules and groups
Special 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.Examples
- If
is the constant group scheme 
over 
, then its corresponding Dieudonné module 
is 
with 
and 
. - For the scheme of p-th roots of unity
, then its corresponding Dieudonné module is 
with 
and 
. - For
, defined as the kernel of the Frobenius 
, the Dieudonné module is 
with 
. - If
is the p-torsion of an elliptic curve over k (with p-torsion in k), then the Dieudonné module depends on whether E is supersingular or not.