This entry is about a section of the text
Let be a morphism of rings. Then we have an adjunction
from the category of -modules to that of -modules where
is called scalar extension and is called scalar restriction.
(Frobenius recognizes p-torsion)
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.
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.
Note that the Frobenius is an endomorphism of a field only if the characteristic of is . In this case it is automatically a monomorphism, since field homomorphisms always are.
However if we pass from rings to schemes, in general it is not true that Frobenius is a monomorphism. The following proposition gives necessary and sufficient conditions for the Frobenius to be a monomorphism in case of formal schemes.
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 .