Abstract
Let \(X\) be a smooth variety over an algebraically closed field of characteristic \(p > 0, Z\) a smooth divisor, and \(j: U=X {\setminus } Z \rightarrow X\) the natural inclusion. We introduce in an axiomatic way the notion of a \(V\)-filtration on unit \(F\)-crystals and prove such axioms determine a unique filtration. It is shown that if \(\mathcal M \) is a tame unit \(F\)-crystal on \(U\), then such a \(V\)-filtration along \(Z\) exists on \(j_*\mathcal M \). The degree zero component of the associated graded module is proven to be the (unipotent) nearby cycles functor of Grothendieck and Deligne under the Emerton–Kisin Riemann–Hilbert correspondence. A few applications to \(\mathbb A ^1\) and gluing are then discussed.
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Acknowledgments
The author benefited from conversations with Sam Gunningham and was greatly assisted by notes from Kari Vilonen about the \(V\)-filtration in characteristic zero. The author would like to thank David Nadler for suggesting to pursue the \(V\)-filtration in positive characteristic in the crystalline setting and his patience and assistance while the author found the correct context for it. He is extremely grateful to Matthew Emerton for his assistance in leading the author through the unit \(F\)-module Riemann–Hilbert correspondence, the suggestion to consider tame ramification and supplying the author with very helpful insights into conducting research in positive characteristic geometry. Lastly, he would also like to thank the referee for providing many useful and thoughtful comments which undoubtedly improved the quality of this manuscript; particularly in the presentation of the proof of 4.2.1.
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Author partially supported by NSF grant DMS-0636646.