The absolute Frobenius morphism is a purely inseparable morphism of degree . Its differential is zero. It preserves products, meaning that for any two schemes and , . Restriction and extension of scalars by Frobenius Suppose that is the structure morphism for an -scheme . The base scheme has a Frobenius morphism FS. Composing with FS results in an -scheme XF called the restriction of scalars by Frobenius. The restriction of scalars is actually a functor, because an -morphism induces an -morphism . For example, consider a ring A of characteristic and a finitely presented algebra over A: The action of A on R is given by: where α is a multi-index. Let . Then is the affine scheme , but its structure morphism , and hence the action of A on R, is different: Because restriction of scalars by Frobenius is simply composition, many properties of are inherited by XF under appropriate hypotheses on the Frobenius morphism. For example, if and SF are both finite type, then so is XF. The extension of scalars by Frobenius is defined to be: The projection onto the factor makes an -scheme. If is not clear from the context, then is denoted by . Like restriction of scalars, extension of scalars is a functor: An -morphism determines an -morphism . As before, consider a ring A and a finitely presented algebra R over A, and again let . Then: A global section of is of the form: where α is a multi-index and every aiα and bi is an element of A. The action of an element c of A on this section is: Consequently, is isomorphic to: where, if: then: A similar description holds for arbitrary A-algebras R. Because extension of scalars is base change, it preserves limits and coproducts. This implies in particular that if has an algebraic structure defined in terms of finite limits (such as being a group scheme), then so does . Furthermore, being a base change means that extension of scalars preserves properties such as being of finite type, finite presentation, separated, affine, and so on. Extension of scalars is well-behaved with respect to base change: Given a morphism , there is a natural isomorphism: Relative Frobenius Let be an -scheme with structure morphism . The relative Frobenius morphism of is the morphism: defined by the universal property of the pullback (see the diagram above): Because the