Dieudonné module explained
In mathematics, a Dieudonné module introduced by, is a module over the non-commutative Dieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms
and
called the
Frobenius and
Verschiebung operators. They are used for studying finite flat commutative group schemes.
of positive characteristic
can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring
,
which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of
. The endomorphisms
and
are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors. Dieudonné and
Pierre Cartier constructed an
antiequivalence of categories between finite commutative group schemes over
of order a power of
and modules over
with finite
-length. The Dieudonné module functor in one direction is given by homomorphisms into the
abelian sheaf
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
, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected
-group schemes correspond to
-modules for which
is nilpotent, and étale group schemes correspond to modules for which
is an isomorphism.
Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Tadao 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, Alexander Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze
-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
Andrew Wiles's work on the
Shimura–Taniyama conjecture.
Dieudonné rings
If
is a perfect field of characteristic
, its ring of
Witt vectors consists of sequences
of elements of
, and has an endomorphism
induced by the Frobenius endomorphism of
, so
. The
Dieudonné ring, often denoted by
or
, is the non-commutative ring over
generated by 2 elements
and
subject to the relations
.
It is a
-graded ring, where the piece of degree
} is a 1-dimensional free module over
, spanned by
if
and by
if
. Some authors define the Dieudonné ring to be the completion of the ring above for the ideal generated by
and
.
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
-group schemes over
.
Examples
is the constant group scheme
over
, then its corresponding Dieudonné module
is
with
and
.
-th roots of unity
, then its corresponding Dieudonné module is
with
and
.
, defined as the kernel of the Frobenius
, the Dieudonné module is
with
.
is the
-torsion of an elliptic curve over
(with
-torsion in
), then the Dieudonné module depends on whether
is
supersingular or not.
Dieudonné–Manin classification theorem
The Dieudonné–Manin classification theorem was proved by and . It describes the structure of Dieudonné modules over an algebraically closed field
up to "isogeny". More precisely, it classifies the finitely generated modules over
, where
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
where
and
are coprime integers with
. The module
has a basis over
of the form
for some element
, and
. The rational number
is called the slope of the module.
The Dieudonné module of a group scheme
If
is a commutative group scheme, its Dieudonné module
is defined to be
, defined as
where
is the formal Witt group scheme and
is the truncated Witt group scheme of Witt vectors of length
.
The Dieudonné module gives antiequivalences between various sorts of commutative group schemes and left modules over the Dieudonné ring
.
- Finite commutative group schemes of
-power order correspond to
modules that have finite length over
.
- Unipotent affine commutative group schemes correspond to
modules that are
-torsion.
-divisible groups correspond to
-modules that are finitely generated free
-modules, at least over perfect fields.
Dieudonné crystal
together with homomorphisms
and
satisfying the relations
(on
),
(on
). Dieudonné crystals were introduced by . They play the same role for classifying algebraic groups over schemes that Dieudonné modules play for classifying algebraic groups over fields.
References