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Suppose $S$ is such that $\mathcal O_S$ is killed by multiplication by $p$. The absolute Frobenius $F: S \to S$ is defined to be be the identity on the underlying points topological space of $S$, with the sheaf map $F^\#: \mathcal O_S \to \mathcal O_S$ given by $x \mapsto x^p$ for any section.

The pullback sheaf is defined to be

$$F^*(\mathcal L) = F^{-1} L \otimes_{F^{-1}\mathcal O_S} \mathcal O_S$$

Where $F^{-1} \mathcal L$ is the sheafification of the presheaf defined by

$$U \mapsto \lim_{F(V)\supset U} \mathcal L (V)$$

But since $F$ is the identity on the underlying space, won't we have $F^{-1} \mathcal L = \mathcal L$ and $F^{-1} \mathcal O_S$ = $\mathcal O_S$, and hence $F^* \mathcal L = \mathcal L$? I can't see how to bring $F^\#$ into the calculation.

I know the intuitive answer is that the transition functions are sent to their p-th powers. But I can't work out the details.

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2 Answers 2

up vote 8 down vote accepted

It is true that $F^{-1}O_S=O_S$, but in the tensor product for $F^*L$, the $O_S$-module structure of the right factor $O_S$ is not given by the identity $O_S\to O_S$, it is the Frobenius $F^{\#}$. It would be less confusing to write $F: T \to S$, and consider the tensor product $F^{-1}L\otimes_{F^{-1}O_S} O_T$.

Further remark on the construction of $F^*L$: you should not use the very definition, but the property that if $S=\mathrm{Spec}(A)$, $T=\mathrm{Spec}(B)$ and $L$ is any quasi-coherent sheaf on $S$ associated to an $A$-module $M$, then $F^*L$ is just the quasi-coherent sheaf on $T$ associated to the $B$-module $M\otimes_A B$.

A proof of $F^*L\simeq L^{\otimes p}$ is as you suggested: consider (the isomorphism class of) the sheaf $L$ as an element of $\mathrm{Pic}(S)=H^1(S, O_S^*)$. Then the effect of taking $F^*$ is given by $F^{\#}: O_S^* \to O_S^*$.

Anothe proof gives directly an isomorphism as follows. Let $A$ be an $\mathbb F_p$-algebra. Denote by $\rho=F^{\#}$ the absolute Frobenius on $A$, and by $A_\rho$ the $A$-algebra whose ring is $A$, but the $A$-algebra structure is given by $\rho : A\to A_\rho$. Similarly, for any $A$-module $M$, let $M_\rho$ be $M$ endowed with the structure of $A$-module via $a*x=\rho(a)x$. This is also an $A_\rho$-module whose structure is exactly that of $M$ as $A$-module. Let $S(M_\rho)$ be the symetric algebra over $A_\rho$ associated to $M_\rho$. We have an $A$-bilinear map $$ \phi: M \times A_\rho \to S(M_\rho), \quad (x, b)\mapsto bx^p.$$ Actually $\phi(ax, b)=b(ax)^p=ba^px^p=\rho(a)\phi(x,b)=a*\phi(x,b)$, and $\phi(x, a*b)=\phi(x, a^pb)=a^p\phi(x,b)=a*\phi(x,b)$. So $\phi$ induces a $A$-linear map $$ M\otimes_A A_\rho \to S(M_\rho), \quad x\otimes b\mapsto bx^p.$$ This map is $A_\rho$-linear too. When $M$ is free of rank one, this induces an isomorphism onto the degree $p$ component of the symetric algebra which is also $M^{\otimes p}$. Continued in the comments.

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(Typing in the main window becomes terrifically slow.) As the $A_\rho$-linear map above is defined canonically, for every quasi-coherent sheaf $\mathcal G$ on $S$, we have a canonical map of quasi-coherent sheaves on $T$: $F^*{\mathcal G}\to S(\mathcal{G})$. When $\mathcal G$ is invertible, this induces an isomorphism onto ${\mathcal G}^{\otimes p}$. –  user18119 Apr 24 '12 at 2:17
    
Thank you so much for such a detailed answer -- you're right it really helps to think in terms of modules rather than sheaves! –  maxymoo Apr 25 '12 at 5:26

Even more simple, and more general:

Let $S$ be a ringed space of $\mathbb{F}_p$-algebras, and let $F : S \to S$ be the absolute Frobenius. Then for every line bundle $\mathcal{L}$ on $S$ we have $F^* \mathcal{L} \cong \mathcal{L}^{\otimes p}$.

Proof. Since $F^*$ is left adjoint to $F_*$ (which could be seen as a definition), for every sheaf of modules $M$ on $S$ we have to prove $\hom(\mathcal{L},F_* M) \cong \hom(\mathcal{L}^{\otimes p},M)$, naturally in $M$. For $\mathcal{L}=\mathcal{O}_S$ this is trivial, both sides identify with $\Gamma(M)$, and in general $\mathcal{L}$ looks locally like this. So we only have to observe that the isomorphisms glue, i.e. that for every isomorphism $\gamma : \mathcal{O}_S \to \mathcal{O}_S$ resp. unit $\gamma \in \Gamma(\mathcal{O}_S)^*$ the diagram

$$\begin{matrix} \hom(\mathcal{O}_S,F_* M) & \cong & \Gamma(F_* M) & = & \Gamma(M) & \cong & \hom(\mathcal{O}_S^{\otimes p},M) \\ \gamma^* \downarrow ~ & & ~ \downarrow \cdot \gamma ~ & & ~ \downarrow \cdot \gamma^p & & ~ \downarrow (\gamma^{\otimes p})^* \\\hom(\mathcal{O}_S,F_* M) & \cong & \Gamma(F_* M) & = & \Gamma(M) & \cong & \hom(\mathcal{O}_S^{\otimes p},M) \end{matrix}$$

commutes, which can be checked directly by the definition of $F$.

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