We give here another characterization of the Frobenius morphism in terms of symmetric products.
Let be a prime number, let be a filed 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.