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**Slope filtrations.**
*(English)*
Zbl 1213.14039

Confluentes Math. 1, No. 1, 1-85 (2009); erratum ibid. 12, No. 1, 147-148 (2020).

Slope filtrations occur in algebraic and analytic geometry, in asymptotic analyis, in ramification theory, in \(p\)-adic theories, in geometry of numbers, etc.. Five basic examples are the Harder-Narasimhan filtration of vector bundles on a smooth projective curve, the Dieudonné-Manin filtration of \(F\)-isocrystals over the \(p\)-adic point, the Turittin-Levelt filtration of formal differential modules, the Hasse-Arf filtration of finite Galois representations of local fields, and the Grayson-Stuhler filtration of Euclidean lattices.

Despite the varieties of their origins, these filtrations share a lot of similar features.

The ‘principle’ is that one can ‘unscrew’ objects \(M\) with slope filtrations in a given additive category according to their Newton polygon, functorially in \(M\). In almost all ‘natural examples’ this principle is enhanced by the combinatorial constraints coming from the fact that the coordinates of the vertices of the Newton polygon are integers.

In this paper, a unified and systematic treatment of slope filtrations is developed, with the aim, as the author writes, of freeing the ‘yoga of stability’ from any ad hoc property underlying the category. This should not only clarify the analogies, but also allow to replace the pervasive adaptations of arguments from one context to another by a single formal argument.

While a large part of this 85 pages text is devoted to the development of an abstract theory, the theory is then also broadly exemplified, notably by the examples mentionded above. The headings of the five chapters are as follows:

0) Introduction

1) General Theory of Slope Filtrations

2) Behaviour of Slope Filtrations with respect to a Tensor Product

3) A Catalogue of Determinantal Slope Filtrations

4) A Catalogue of \(\otimes\)-bounded Slope Filtrations

5) Variation of Newton Polygons in Families

Appendix A: Pseudo \(\otimes\)-functors and Rigidity

Despite the varieties of their origins, these filtrations share a lot of similar features.

The ‘principle’ is that one can ‘unscrew’ objects \(M\) with slope filtrations in a given additive category according to their Newton polygon, functorially in \(M\). In almost all ‘natural examples’ this principle is enhanced by the combinatorial constraints coming from the fact that the coordinates of the vertices of the Newton polygon are integers.

In this paper, a unified and systematic treatment of slope filtrations is developed, with the aim, as the author writes, of freeing the ‘yoga of stability’ from any ad hoc property underlying the category. This should not only clarify the analogies, but also allow to replace the pervasive adaptations of arguments from one context to another by a single formal argument.

While a large part of this 85 pages text is devoted to the development of an abstract theory, the theory is then also broadly exemplified, notably by the examples mentionded above. The headings of the five chapters are as follows:

0) Introduction

1) General Theory of Slope Filtrations

2) Behaviour of Slope Filtrations with respect to a Tensor Product

3) A Catalogue of Determinantal Slope Filtrations

4) A Catalogue of \(\otimes\)-bounded Slope Filtrations

5) Variation of Newton Polygons in Families

Appendix A: Pseudo \(\otimes\)-functors and Rigidity

Reviewer: Elmar Große-Klönne (Berlin)

### MSC:

14F30 | \(p\)-adic cohomology, crystalline cohomology |

11G25 | Varieties over finite and local fields |

12H25 | \(p\)-adic differential equations |

14G22 | Rigid analytic geometry |