There are several different ways to define the Frobenius morphism for a
scheme. The most fundamental is the absolute Frobenius morphism. However, the absolute Frobenius morphism behaves poorly in the relative situation because it pays no attention to the base scheme. There are several different ways of adapting the Frobenius morphism to the relative situation, each of which is useful in certain situations. . The morphism is relative Frobenius.
The absolute Frobenius morphism Suppose that is a scheme of characteristic . Choose an open affine subset of . The ring is an -algebra, so it admits a Frobenius endomorphism. If is an open affine subset of , then by the naturality of Frobenius, the Frobenius morphism on , when restricted to , is the Frobenius morphism on . Consequently, the Frobenius morphism glues to give an endomorphism of . This endomorphism is called the
absolute Frobenius morphism of , denoted . By definition, it is a homeomorphism of with itself. The absolute Frobenius morphism is a natural transformation from the identity functor on the category of -schemes to itself. If is an -scheme and the Frobenius morphism of is the identity, then the absolute Frobenius morphism is a morphism of -schemes. In general, however, it is not. For example, consider the ring A = \mathbf{F}_{p^2}. Let and both equal with the structure map being the identity. The Frobenius morphism on sends to . It is not a morphism of \mathbf{F}_{p^2}-algebras. If it were, then multiplying by an element in \mathbf{F}_{p^2} would commute with applying the Frobenius endomorphism. But this is not true because: : b \cdot a = ba \neq F(b) \cdot a = b^p a. The former is the action of in the \mathbf{F}_{p^2}-algebra structure that begins with, and the latter is the action of \mathbf{F}_{p^2} induced by Frobenius. Consequently, the Frobenius morphism on is not a morphism of \mathbf{F}_{p^2}-schemes. 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 . Composing with 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 of characteristic and a finitely presented algebra over : :R = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m). The action of on is given by: :c \cdot \sum a_\alpha X^\alpha = \sum c a_\alpha X^\alpha, where α is a multi-index. Let . Then is the affine scheme , but its structure morphism , and hence the action of on , is different: : c \cdot \sum a_\alpha X^\alpha = \sum F(c) a_\alpha X^\alpha = \sum c^p a_\alpha X^\alpha. Because restriction of scalars by Frobenius is simply composition, many properties of are inherited by under appropriate hypotheses on the Frobenius morphism. For example, if and
SF are both finite type, then so is . The
extension of scalars by Frobenius is defined to be: : X^{(p)} = X \times_S S_F. 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 and a finitely presented algebra over , and again let . Then: : X^{(p)} = \operatorname{Spec} R \otimes_A A_F. A global section of is of the form: : \sum_i \left(\sum_\alpha a_{i\alpha} X^\alpha\right) \otimes b_i = \sum_i \sum_\alpha X^\alpha \otimes a_{i\alpha}^p b_i, where is a multi-index and every and is an element of
A. The action of an element
c of
A on this section is: : c \cdot \sum_i \left (\sum_\alpha a_{i\alpha} X^\alpha\right) \otimes b_i = \sum_i \left(\sum_\alpha a_{i\alpha} X^\alpha\right) \otimes b_i c. Consequently, is isomorphic to: : \operatorname{Spec} A[X_1, \ldots, X_n] / \left (f_1^{(p)}, \ldots, f_m^{(p)} \right ), where, if: : f_j = \sum_\beta f_{j\beta} X^\beta, then: : f_j^{(p)} = \sum_\beta f_{j\beta}^p X^\beta. A similar description holds for arbitrary -algebras . 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: :X^{(p/S)} \times_S S' \cong (X \times_S S')^{(p/S')}.
Relative Frobenius Let be an -scheme with structure morphism . The
relative Frobenius morphism of is the morphism: : F_{X/S} : X \to X^{(p)} defined by the
universal property of the
pullback (see the diagram above): : F_{X/S} = (F_X, \varphi). Because the absolute Frobenius morphism is natural, the relative Frobenius morphism is a morphism of -schemes. Consider, for example, the -algebra: : R = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m). We have: : R^{(p)} = A[X_1, \ldots, X_n] / (f_1^{(p)}, \ldots, f_m^{(p)}). The relative Frobenius morphism is the homomorphism defined by: : \sum_i \sum_\alpha X^\alpha \otimes a_{i\alpha} \mapsto \sum_i \sum_\alpha a_{i\alpha}X^{p\alpha}. Relative Frobenius is compatible with base change in the sense that, under the natural isomorphism of and , we have: : F_{X / S} \times 1_{S'} = F_{X \times_S S' / S'}. 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 . is unramified over
if and only if is unramified and if and only if is a monomorphism. is étale over if and only if is étale and if and only if is an isomorphism.
Arithmetic Frobenius The
arithmetic Frobenius morphism of an -scheme is a morphism: : F^a_{X/S} : X^{(p)} \to X \times_S S \cong X defined by: : F^a_{X/S} = 1_X \times F_S. That is, it is the base change of by . Again, if: : R = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m), : R^{(p)} = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m) \otimes_A A_F, then the arithmetic Frobenius is the homomorphism: : \sum_i \left(\sum_\alpha a_{i\alpha} X^\alpha\right) \otimes b_i \mapsto \sum_i \sum_\alpha a_{i\alpha} b_i^p X^\alpha. If we rewrite as: : R^{(p)} = A[X_1, \ldots, X_n] / \left (f_1^{(p)}, \ldots, f_m^{(p)} \right ), then this homomorphism is: : \sum a_\alpha X^\alpha \mapsto \sum a_\alpha^p X^\alpha.
Geometric Frobenius Assume that the absolute Frobenius morphism of is invertible with inverse F_S^{-1}. Let S_{F^{-1}} denote the -scheme F_S^{-1} : S \to S. Then there is an extension of scalars of by F_S^{-1}: : X^{(1/p)} = X \times_S S_{F^{-1}}. If: : R = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m), then extending scalars by F_S^{-1} gives: : R^{(1/p)} = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m) \otimes_A A_{F^{-1}}. If: : f_j = \sum_\beta f_{j\beta} X^\beta, then we write: : f_j^{(1/p)} = \sum_\beta f_{j\beta}^{1/p} X^\beta, and then there is an isomorphism: : R^{(1/p)} \cong A[X_1, \ldots, X_n] / (f_1^{(1/p)}, \ldots, f_m^{(1/p)}). The
geometric Frobenius morphism of an -scheme is a morphism: : F^g_{X/S} : X^{(1/p)} \to X \times_S S \cong X defined by: : F^g_{X/S} = 1_X \times F_S^{-1}. It is the base change of F_S^{-1} by . Continuing our example of and above, geometric Frobenius is defined to be: : \sum_i \left(\sum_\alpha a_{i\alpha} X^\alpha\right) \otimes b_i \mapsto \sum_i \sum_\alpha a_{i\alpha} b_i^{1/p} X^\alpha. After rewriting in terms of \{f_j^{(1/p)}\}, geometric Frobenius is: : \sum a_\alpha X^\alpha \mapsto \sum a_\alpha^{1/p} X^\alpha.
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 on . Consider the set of -points . This set comes with a Galois action: Each such point corresponds to a homomorphism from the structure sheaf to , which factors via , the residue field at , and the action of Frobenius on is the application of the Frobenius morphism to the residue field. This Galois action agrees with the action of arithmetic Frobenius: The composite morphism : \mathcal{O}_X \to k(x) \xrightarrow{\overset{}F} k(x) is the same as the composite morphism: : \mathcal{O}_X \xrightarrow{\overset{}F^a_{X/S}} \mathcal{O}_X \to k(x) by the definition of the arithmetic Frobenius. Consequently, arithmetic Frobenius explicitly exhibits the action of the Galois group on points as an endomorphism of . == Frobenius for local fields ==