absolute Frobenius morphism is natural, the relative Frobenius morphism is a morphism of -schemes. Consider, for example, the A-algebra: We have: The relative Frobenius morphism is the homomorphism defined by: Relative Frobenius is compatible with base change in the sense that, under the natural isomorphism of and , we have: Relative Frobenius is a universal homeomorphism. If is an open immersion, then it is the identity. If is a closed immersion determined by an ideal sheaf I of , then is determined by the ideal sheaf and relative Frobenius is the augmentation map . X is unramified over if and only if FX/S is unramified and if and only if FX/S is a monomorphism. X is étale over if and only if FX/S is étale and if and only if FX/S is an isomorphism. Arithmetic Frobenius The arithmetic Frobenius morphism of an -scheme is a morphism: defined by: That is, it is the base change of FS by 1X. Again, if: then the arithmetic Frobenius is the homomorphism: If we rewrite as: then this homomorphism is: Geometric Frobenius Assume that the absolute Frobenius morphism of is invertible with inverse . Let denote the -scheme . Then there is an extension of scalars of by : If: then extending scalars by gives: If: then we write: and then there is an isomorphism: The geometric Frobenius morphism of an -scheme is a morphism: defined by: It is the base change of by . Continuing our example of A and R above, geometric Frobenius is defined to be: After rewriting R in terms of , geometric Frobenius is: Arithmetic and geometric Frobenius as Galois actions Suppose that the Frobenius morphism of is an isomorphism. Then it generates a subgroup of the automorphism group of . If is the spectrum of a finite field, then its automorphism group is the Galois group of the field over the prime field, and the Frobenius morphism and its inverse are both generators of the automorphism group. In addition, and may be identified with . The arithmetic and geometric Frobenius morphisms are then endomorphisms of , and so they lead to an action of the Galois group of k on X. Consider the set of K-points . This set comes with a Galois action: Each such point x corresponds to a homomorphism from the structure sheaf to K, which factors via k(x), the residue field at x, and