Let $X$ be a scheme over a finite field $\mathbb{F}\_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}\_X$-module, then there is a natural isomorphism $F^\*(\mathcal{L}) \cong \mathcal{L}^{\otimes q}$.



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> Is there also a general formula for $F^\*(\mathcal{M})$ if $\mathcal{M}$ is a locally free $\mathcal{O}\_X$-module of given rank $d$?
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Of course, we have $F^\*(\mathcal{M}) = \mathcal{M} \otimes\_{\mathcal{O}\_X} F\_\*(\mathcal{O}\_X)$ (since $F$ is affine), but I would like to have a formula which is independent of the quasi-coherent algebra $F\_\*(\mathcal{O}\_X)$ (unless you can describe this algebra via generators and relations), but only depends on $\mathcal{M}$ and uses the usual operations on quasi-coherent modules ($\otimes$, $\ker$, $\mathrm{coker}$, $\oplus$, $\mathrm{Sym}^n$, $\Lambda^n$, $\dotsc$).


For $q=2$ I have found that $F^\*(\mathcal{M})$ is the kernel of the homomorphism $\mathrm{Sym}^2(\mathcal{M}) \twoheadrightarrow \Lambda^2(\mathcal{M})$, $m \cdot n \mapsto m \wedge n$. In general, there is an embedding$$F^\*(\mathcal{M}) \to \mathrm{Sym}^q(\mathcal{M})$$ which maps $m \otimes 1 \mapsto m^q$.


My question may be also phrased as follows: On the $K$-theory of $X$, $F^\*$ induces the Adams operation $\psi^q$. Hence, I ask for a bundle representative of the Adams operation $\psi^q$. (Not just a formal difference of bundles.)



Pullback along Frobenius morphism
Consider the $p$-th tensor power $\mathcal{M}^{\otimes p}$. The group $\mathbb{Z}/p\mathbb{Z}$ acts by cyclic permutations on it. Denote its generator by $\sigma$. There is a map from coinvariants to invariants $$(\mathcal{M}^{\otimes p})\_{\sigma}\xrightarrow{1+\sigma+\dots+ \sigma^{p-1}}(\mathcal{M}^{\otimes p})^{\sigma}$$ Its kernel is canonically isomorphic to $F^\*\mathcal{M}$ via the map $s\mapsto s^{\otimes p}$. This is proven in Lemma 6.9 here:<http://arxiv.org/pdf/1509.08784v1.pdf> for the affine case and the general case follows by functoriality.


Edit: By the way, its cokernel is also isomorphic to $F^\*\mathcal{M}$.


