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Let be a perfect field of prime characteristic .
Let denote the codirected system of affine commutative unipotent? -groups
The Witt ring operates on as follows:
Let denote the Frobenius morphism , let . This Frobenius is bijective since is perfect. Let , let , . We define
where is the image of in , and , the product of and . By this definition becomes a -module, and is a homomorphism of -modules since we have
For any the Dieudonné module of is defined to be the -module
or- equivalently- where denotes the category of codirected diagrams in as described above.
is a contravariant functor from affine commutative unipotent -groups to the category of -modules.
This construction commutes with automorphisms of . In particular it commutes with the morphism .
For a -module , define .
As a group , but the external law is .
If , then is a morphism and hence a map
(…) There is an isomorphism .
The Frobenius morphism and the Verschiebung morphisminduce morphisms of modules. and .
The translation morphism is a monomorphism and the maps are injective.
can be identified with a submodule of , namely and we say that an element of is killed by a power of .
Let be the (non-commutative) ring generated by and two elements and subject to the relations
Any element of can be written uniquely as a finite sum
If , then has a canonical structure of a left -module. If is a perfect extension of , then there is a canonical map of -modules
Note that and the left hand side can also be written .
The functor induces an (contravariant) equivalence
between and the category of all -modules of -torsion.
For any perfect extension \of we have that
is an isomorphism. Moreover
is algebraic iff is a finitely generated -module.
is finite iff is a -module of finite length.