Frobenius morphism of fields
Suppose is a field of positive characteristic . The Frobenius morphism is an endomorphism of the field defined by
Notice that this is indeed a homomorphism of fields: the identity evidently holds for all and the characteristic of the field is used to show .
Frobenius is always injective. Note that the Frobenius morphism of schemes (see below) is not always a monomorphism.
The image of Frobenius is the set of elements of with a -th root and is sometimes denoted .
Frobenius is surjective if and only if is perfect.
Frobenius morphism of schemes
In terms of schemes as locally ringed spaces
Suppose is an -scheme where is a scheme over . The absolute Frobenius is the map which is the identity on the topological space and on the structure sheaves is the -th power map. This is not a map of -schemes in general since it doesn’t respect the structure of as an -scheme, i.e. the diagram:
so in order for the map to be an -scheme morphism, must be the identity on , i.e. .
Now we can form the fiber product using this square: . By the universal property of pullbacks there is a map so that the composition is . This is called the relative Frobenius. By construction the relative Frobenius is a map of -schemes.
For the purposes below will be a perfect field of characteristic >.
is smooth over if and only if is a vector bundle, i.e. is a free -module of rank . One can study singularities of by studying properties of .
If is smooth and proper over , the sequence is exact and if it splits then has a lifting to .
In terms of schemes as sheaves on
Let be a prime number, let be a field of characteristic . For a -ring we define
The -ring obtained from by scalar restriction along is denoted by .
The -ring obtained from by scalar extension along is denoted by .
There are -ring morphisms and .
For a -functor we define which satisfies . The Frobenius morphism for is the transformation of -functors defined by
If is a -scheme is a -scheme, too.
Since the completion functor commutes with the above constructions the Frobenius morphism can be defined for formal k-schemes, too.
In terms of symmetric products
We give here another characterization of the Frobenius morphism in terms of symmetric products.
Let be a prime number, let be a field of characteristic , let be a -vector space, let denote the -fold tensor power of , let denote the subspace of symmetric tensors. Then we have the symmetrization operator
end the linear map
then the map is bijective and we define by
If is a -ring we have that is a -ring and is a -ring morphism.
If is a ring spectrum we abbreviate and the following diagram is commutative.
The Frobenius as a morphism (natural transformation) of (affine) group schemes is one operation among other (related) operations of interest:
For a more detailed account of the relationship of Frobenius-, Verschiebung-? and homothety morphism? see Hazewinkel
Frobenius morphism of -rings
If is a -ring spectrum we have and .
If is a finite field we have however will not equal in general.
If is a field extension we have .