Purity for overconvergence

Abstract

Let X → X be an open immersion of smooth varieties over a field of characteristic p > 0 such that the complement is a simple normal crossing divisor and Z ⊆ Z ⊆ X closed subschemes of codimension at least 2. In this paper, we prove that the canonical restriction functor between the categories of overconvergent F-iso-is an equivalence of categories. We also give an application of our result to the equivalence of certain categories. Mathematics Subject Classification (2010)12H25 · 14F35 IntroductionLet X be a regular scheme and Z ⊆ X a closed subscheme of codimension at least 2. Then, by the famous Zariski-Nagata purity, any locally constant constructible sheaf on (X \Z ) et extends uniquely to a locally constant constructible sheaf on X et . As a p-adic analog of this fact, Kedlaya proved in [11, 5.3.3] the following result on the purity for overconvergent isocrystals. Let k be a field of characteristic p > 0, X → X an open immersion of k-varieties with X smooth and Z ⊆ X a closed subscheme of X of codimension at least 2 such that X \Z is dense in X . Then, the restriction functor between the categories of overconvergent isocrystals Isoc † (X, X ) −→ Isoc † (X \Z , X ) is an equivalence of categories.

Log-decay $F$-isocrystals on higher dimensional varieties

“…The proof of Drinfeld and Kedlaya in [4] can be summarized as follows: first they prove that if U ⊂ X is a dense open subscheme, the restriction functor from convergent F -isocrystals on X to F -isocrystals on U is fully faithful. This builds upon several other difficult fully faithful results due to Kedlaya and Shiho (see [8], [11,Theorem 5.1], and [15]). Let M be an F -isocrystal on X.…”

Langlands program for p-adic coefficients and the petits camarades conjecture

“…Assume there exists a smooth compactification X ֒→ X such that the complement is a simple normal crossing divisor. Then by the same argument as [Cr1,4.13], using [Sh,Thm 4.3] and [KL,Theorem 2] instead of [Cr1,4.12] and the class field theory, we get that (i) holds in this case.…”

Drinfeld's lemma for F-isocrystals, II: Tannakian approach

“…We may shrink each and, in particular, we may assume that admits a smooth compactification such that the complement is a simple normal crossing divisor. Using [Shi11b, Theorem 4.3], [KL81, Theorem 2], and the same argument of [Cre87, Corollary 4.13], one can show that some power is geometrically trivial, i.e. its underlying overconvergent isocrystal is trivial.…”