Remarks on F-spans

We prove a version of Frobenius descent with applications to the theory of F-crystals and F-spans. Let $X/S$ be a smooth morphism of schemes in characteristic $p$, let $F_{X/S} \colon X/S \to X’/S$ be the relative Frobenius morphism, and let $F \colon Z/T \to Z’/T$ be a lifting of $F_{X/S}$, where $Z,Z’$, and $T$ are $p$-torsion free $p$-adic formal schemes. If $(E’,\nabla ’)$ is a $p$-torsion free coherent sheaf with integrable and quasi-nilpotent connection on $Z’$, we show that any submodule of $F^*(E’)$ which is invariant under the induced connection descends to $E’$. As a consequence, we show that if $\Phi \colon F^*(E’, \nabla ’) \to (E,\nabla )$ is an F-span, then the Mazur–Nygaard filtration on $F_{X/S}^*(E’_{X’})$ descends naturally to a filtration on $E’_{X’}$ which satisfies Griffiths transversality. This generalizes an earlier result of the author, which required that the Frobenius lift $F$ be of a special form. We also investigate how the Mazur–Nygaard filtration depends on the lifting $F$.