In number theory, Galois theory and arithmetic geometry in prime characteristic , the Frobenius morphism is the endomorphism acting on algebras, function algebras, structure sheaves etc., which takes each ring/algebra-element to its th power
It is precisely in positive characteristic that this operation is indeed an algebra homomorphism.
The Frobenius map is the shadow of the power operations in multiplicative cohomology theory/higher algebra (Lurie, remark 2.2.7).
The presence of the Frobenius endomorphism in characteristic is a fundamental property in arithmetic geometry that controls many of its deep aspects. Notably zeta functions are typically expressed in terms of the action of the Frobenius endomorphisms on cohomology groups and so it features prominently for instance in the Weil conjectures.
In Borger's absolute geometry lifts of Frobenius endomorphisms through base change for all primes at once – in the sense of Lambda-ring structure – is interpreted as encoding descent data from traditional arithmetic geometry over Spec(Z) down to the “absolute” geometry over “F1”. See at Borger’s absolute geometry – Motivation for more on this.
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Let be 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 .
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 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.
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.
As elements of the Galois group
Some powers of the Frobenius morphism canonically induce elements in the Galois group (…)
Review of the standard story is for instance in (Snyder 02, section 1.5). Further developments include (Dokchitser-Dokchitser 10)
If is a finite field,then is generated by the Frobenius map (e.g. Snyder 02, lemma 1.5.10).
See also at Artin L-function.
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 .
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
Lecture notes include
Further discussion of the relation to the Galois group includes
Noah Snyder, section 1.5 of Artin L-Functions: A Historical Approach, 2002 (pdf)
Tim Dokchitser, Vladimir Dokchitser, Identifying Frobenius elements in Galois groups (arXiv:1009.5388)
Discussion in the context of power operations on E-infinity rings is in