The Harder-Narasimhan filtration on finite flat group schemes. (La filtration de Harder-Narasimhan des schémas en groupes finis et plats.) (English) Zbl 1199.14015

The purpose of this paper is to introduce filtrations on finite flat group schemes over unequal characteristic valuation rings which have formal properties completely parallel to those of the well-known Harder-Narasimhan filtrations on vector bundles on smooth projective curves. More generally, such filtrations are even defined and studied for families of finite flat group schemes parametrized by formal schemes.
For a smooth connected projective curve \(X\) over a field, the category \({\mathcal C}\) of vector bundles on \(X\) is exact and endowed with the two \({\mathbb Z}\)-valued functions \(\text{rank}\) and \(\text{deg}\) which are additive on short exact sequences. There is a ‘generic fibre’ functor from \({\mathcal C}\) to the category of finite dimensional vector spaces over the function field of \(X\), and for a given vector bundle on \(X\) this functor induces a bijection between the respective subobjects. The degree function \(\text{deg}\) grows under modifications (morphisms of vector bundles which are generic isomorphisms) and detects isomorphisms (among morphisms of vector bundles which are generic isomorphisms). From these formal properties the Harder-Narasimhan filtration on vector bundles on \(X\) is defined using the function \[ \mu=\frac{\text{deg}}{\text{rank}}. \] Let \(p\) be a prime number, \(K\) a complete valued field of characteristic \(0\) whose valuation \(v\) takes values in \({\mathbb R}\) and extends the \(p\)-adic valuation. Let now \({\mathcal C}\) denote the category of finite \(p\)-power order and flat commutative group schemes over \({\mathcal O}_K\) (\(=\) the valuation ring of \(K\)). One has the following two \({\mathbb R}\)-valued functions on \({\mathcal C}\): firstly, the height function \(\text{ht}\) (for \(G\in{\mathcal C}\) we have \(|G|=p^{\text{ht}(G)}\)); secondly, the degree function \(\text{deg}\) (for \(G\in{\mathcal C}\) write the conormal sheaf \(\omega_G\) as \(\omega_G=\bigoplus_i{\mathcal O}_K/a_i{\mathcal O}_K\), then \(\text{deg}(G)=\sum_i v(a_i)\): the ‘discriminant’ of \(G\)). It is shown that \({\mathcal C}\), together with the functions \(\text{ht}\) and \(\text{deg}\) satisfies the formal properties analogous to those of the category of vector bundles on a smooth connected projective curve just recalled. Therefore, based on the function \[ \mu=\frac{\text{deg}}{\text{ht}} \] one can now define a theory of Harder-Narasimhan filtrations on objects of \({\mathcal C}\) faithfully following the prototype of vector bundles on a smooth connected projective curve.
One has the corresponding Harder-Narasimhan polygons, a notion of semistability. The semistable \(G\)’s form an abelian category.
Further topics are: groups with additional structures, (kernels of) isogenies of formal \(p\)-divisible groups of dimension \(1\), families of Harder-Narasimhan filtrations over formal schemes (e.g. (semi)continuity properties of the Harder-Narasimhan polygon with respect to the topology of the generic fibre, viewed as a Berkovich analytic space), comparison with Hodge polygons, Hodge-Tate map, more general base schemes.


14L05 Formal groups, \(p\)-divisible groups
14L15 Group schemes
Full Text: DOI


[1] DOI: 10.1353/ajm.2002.0026 · Zbl 1084.11064 · doi:10.1353/ajm.2002.0026
[2] DOI: 10.1007/BF02712916 · Zbl 0804.32019 · doi:10.1007/BF02712916
[3] DOI: 10.1007/s002220050078 · Zbl 0852.14002 · doi:10.1007/s002220050078
[4] DOI: 10.1007/BF01444889 · Zbl 0808.14017 · doi:10.1007/BF01444889
[5] DOI: 10.1007/BF01445112 · Zbl 0808.14018 · doi:10.1007/BF01445112
[6] Elkik R., Ann. Sci. E’c. Norm. 6 (4) pp 553– (1974)
[7] Fargues L., Progr. Math. 262 pp 1– (2008)
[8] DOI: 10.1215/S0012-7094-07-14033-X · Zbl 1136.14013 · doi:10.1215/S0012-7094-07-14033-X
[9] Grothendieck A., Inst. Hautes E’t. Sci. Publ. Math. 20 pp 259– (1964)
[10] DOI: 10.1007/BF01357141 · Zbl 0324.14006 · doi:10.1007/BF01357141
[11] Illusie Luc, Astérisque pp 223– (1994)
[12] Knudsen Finn Faye, Math. Scand. 39 (1) pp 19– (1976)
[13] DOI: 10.2307/2946529 · Zbl 0804.14019 · doi:10.2307/2946529
[14] Raynaud M., Bull. Soc. Math. France 102 pp 241– (1974)
[15] Raynaud M., Astérisque 127 pp 199– (1985)
[16] Stephen, Compos. Math. 35 (2) pp 163– (1977)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.