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Full faithfulness for overconvergent $$F$$-isocrystals. (English) Zbl 1087.14018
Adolphson, Alan (ed.) et al., Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter (ISBN 3-11-017478-2/hbk). 819-835 (2004).
Let $$k$$ be a perfect field of characteristic $$p$$, $$K$$ be the fraction field of a complete mixed characteristics discrete valuation ring with residue field $$k$$, and $$X$$ a smooth $$k$$-scheme of finite type. For an integer $$a \geq 1$$, we denote by $$F^a \mathrm{-Isoc}(X/K)$$ and $$F^a \mathrm{-Isoc}^{\dagger}(X/K)$$ the categories of convergent and overconvergent $$F^a$$-isocrystals on $$X$$. There is a natural forgetful functor $$j^*: F^a \mathrm{-Isoc}^{\dagger}(X/K) \rightarrow F^a \mathrm{-Isoc}(X/K)$$ and the main result of this paper (theorem 1.1) is that this functor is fully faithful. This result had been conjectured by N. Tsuzuki [Duke Math. J. 111, No. 3, 385–418 (2002; Zbl 1055.14022)] and proved, in the unit-root case, by him (ibid.) if $$X$$ has a smooth compactification and by J.-Y. Etesse [Ann. Sci. Ec. Norm. Sup., IV. Sér. 35, No. 4, 575–603 (2002; Zbl 1060.14028)] for general $$X$$. The method of proof of theorem 1.1 is to reduce the result to a local assertion: a full faithfulness result for $$(\sigma,\nabla)$$-modules over convergent and overconvergent rings of power series (theorem 5.1). The proof of that result uses the $$p$$-adic local monodromy theorem. The author also proves that finite projective modules over certain rings of multivariable overconvergent power series are free (theorem 6.6). The paper is very clear and its structure reflects well the different steps of the proof.
For the entire collection see [Zbl 1047.14001].

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 11G25 Varieties over finite and local fields 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
##### Keywords:
overconvergent; isocrystal; monodromy
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