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Slope filtrations in families. (English) Zbl 1285.14023
This paper is of interest to those studying number theory. It studies how certain invariants (HN-polygons) of $$\phi$$-modules over the Robba ring behave in families.
Let $$K$$ be a field equipped with a discrete valuation. Suppose that $$K$$ is complete of characteristic zero and the residue field of the ring of integers is of characteristic $$p >0$$. Let $$R$$ be a ring equipped with an endomorphism $$\phi$$. A $$\phi$$-module over $$R$$ is a pair of a finite free $$R$$-module $$M$$ and an $$R$$-linear isomorphism $$\theta: \phi^*M \rightarrow M$$. An example of interest to this paper is to take $$R=\mathcal{R}_{A_K}$$ to be the Robba ring of an affinoid algebra $$A_K$$ and $$\phi$$ a lift of the relative Frobenius.
The structure morphism $$\theta$$ of a $$\phi$$-module $$M$$ and the valuation on $$K$$ can be used to define the degree of $$M$$. The slope of $$M$$ is defined as $$\mu(M) = \deg(M)/\mathrm{rank}(M)$$. K. S. Kedlaya [Astérisque 319, 259–301 (2008; Zbl 1168.11053)] has shown that $$M$$ has a unique Harder-Narasimhan (HN) filtration $$0 = M_0 \subset M_1 \subset \dots \subset M_l =M$$ by saturated $$\phi$$-submodules. The slopes of this filtration determine a Newton polygon called the HN-polygon.
The paper under review studies how these HN-filtrations and their associated HN-polygons behave in families. It studies the fiberwise behavior of the HN-filtration/HN-polygons of $$\phi$$-modules over $$\mathcal{R}_{A_K}$$. The paper has four main results/applications. For brevity, only one will be fully stated. For any $$\phi$$-module $$M_A$$ over $$\mathcal{R}_{A_K}$$ and for any $$x \in M(A)$$ there is a Weierstrass subdomain $$M(B)$$ of $$M(A)$$ such that the subset of points of $$M(B)$$ where the HN-polygons agree with the HN-polygon at $$x$$ is a Zariski closed set $$M(C)$$. Moreover, the HN-filtration of the fiber $$(M_A)_x$$ at $$x$$ lifts to a slope filtration on the module $$\widetilde{M_C} = M_{A} \otimes_{\mathcal{R}_{A_K}} (C \widehat{\otimes}_{\mathbb{Q}_p} \widetilde{\mathcal{R}}_L)$$ where $$L$$ is an admissible extension of $$K$$ with strongly difference-closed residue field. The other main theorems are variations on this theorem when the hypothesis is that the fiber $$(M_A)_x$$ is pure of slope $$s$$, when $$K$$ is assumed to be a finite unramified extension of $$\mathbb{Q}_p$$, or where the conclusion is that the HN-polygon at $$y$$ lies above the HN-polygon at $$x$$.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 11S20 Galois theory
##### Keywords:
slope filtration; $$\phi$$-modules
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##### References:
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