“The mathematics are usually considered as being the very antipodes of Poesy. Yet Mathesis and Poesy are of the closest kindred, for they are both works of the imagination.”
Thomas Hill
The Frobenius endomorphism of schemes confuses the hell out of me. There’s the absolute Frobenius, the relative Frobenius, the arithmetic Frobenius, the geometric Frobenius… It’s a huge mess of concepts. So, I think it’ll probably benefit me to write an article elucidating these ideas. Our task is to generalize the Frobenius endomorphism \(\mathrm{Frob}_R:R\rightarrow R\) defined by \(x\mapsto x^p\) for a commutative ring \(R\) of characteristic \(p\) (recall \(\mathrm{Frob}_{\mathbb F_{p^n}}\) generates the Galois group \(\mathrm{Gal}(\mathbb F_{p^n}/\mathbb F_p)\)), to the more general case of an endomorphism of a scheme of characteristic \(p\). We begin by defining the absolute Frobenius, and show that it has some undesirable properties, which prompts us to make modifications.
Definition. Let \(X\) be a scheme of characteristic \(p>0\). The absolute Frobenius endomorphism \(\mathrm{Frob}_X:X\rightarrow X\) (sometimes written as \(F_X\)) is defined as the identity on the topological space, and \(\mathrm{Frob}_X^{\flat }\) is given by the usual Frobenius \(\mathrm{Frob}_A\) for each open affine \(\mathrm{Spec}(A)=U\subseteq X\).
This seems to be a natural generalization, but we run into the problem that for an \(S\)-scheme \(f:X\rightarrow S\), we have \(\mathrm{Frob}_X\) is in general not an \(S\)-scheme morphism. For example, take \(X=S=\mathrm{Spec}(A)\) where \(A=\mathbb F_{p^2}\) with identity as the structure map. Note that \(\mathrm{Frob}_{A}\) is not a \(A\)-algebra morphism, hence \(\mathrm{Frob}_X\) is not an \(S\)-morphism. Thus, this motivates us to define a relative variant. Let \(f:X\rightarrow S\) be an \(S\)-scheme, consider the diagram where \(X^{(p)}=X\times_S S\) is the base change of \(X\) by the Frobenius (as in Cartesian square in the diagram), and \(F_{X/S}=\mathrm{Frob}_{X/S}:X\rightarrow X^{(p)}\) is the unique morphism for the diagram to commute i.e. defined by the universal property, called the relative Frobenius of \(X\) over \(S\). For example, take \(A\) a ring of characteristic \(p\), \(R\) a finitely presented algebra over \(A\), and \(X=\mathrm{Spec}R\), then \(X^{(p)}=\mathrm{Spec}(R\otimes_{A,F_A} A)\), which is an extension of scalars. Therefore if \(R=A[X_1,\dots,X_n]/(f_1,\dots,f_m)\) then \(R^{(p)}\cong A[X_1,\dots,X_n]/(f_1^{(p)},\dots,f_m^{(p)})\) where \(f^{(p)}=\sum_\alpha s_\alpha^p X^\alpha\) for \(f=\sum_\alpha s_\alpha X^\alpha\), and \(X=\mathrm{Spec}(R^{(p)})\). The morphism \(\sigma_X^*\) is induced by the endomorphism \(f \mapsto f^{(p)}\) on \(A[X_1,\dots,X_n]\), and the morphism \(F_{X/S}^*=\mathrm{Frob}_{X/S}^*\) is induced by the endmorphism of \(A\)-algebras \(X_i\mapsto X_i^p\) on \(A[X_1,\dots,X]\).
Theorem. The relative Frobenius is compatible with base change in the sense that
\[\mathrm{Frob}_{X/S}\times_S \mathbf{1}_T = \mathrm{Frob}_{(X\times_S T)/T}\] identifying \(X^{(p/S)}\times_S T\cong (X\times_S T)^{(p/T)}\) canonically.
There is also the arithmetic and geometric Frobenius, which are defined as base changes of the absolute Frobenius. The arithmetic Frobenius \(\mathrm{Frob}_{X / S}^a: X^{(p)} \rightarrow X \times_S S \cong X\) is the base change \(\mathrm{Frob}_{X / S}^a=\mathbf{1}_X \times_S \mathrm{Frob}_S\). Assume that the absolute Frobenius \(\mathrm{Frob}_S\) is invertible, then the geometric Frobenius \(\mathrm{Frob}_{X / S}^g: X^{(1 / p)} \rightarrow X \times_S S \cong X\) is the base change \(\mathrm{Frob}^g_{X/S}=\mathbf{1}_X\times_S \mathrm{Frob}_S^{-1}\).
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