Slope filtrations in families.

*(English)*Zbl 1285.14023This 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 \(x\).

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 \(x\).

Reviewer: Theodore J. Stadnik Jr. (Berkeley)

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\textit{R. Liu}, J. Inst. Math. Jussieu 12, No. 2, 249--296 (2013; Zbl 1285.14023)

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