Ramification estimate for Fontaine-Laffaille Galois modules. (English) Zbl 1322.11122

Abrashkin, Victor

J. Algebra 427, 319-328 (2015).

Let \(W(k)\) be the ring of Witt vectors with coefficients in a perfect field \(k\) of characteristic \(p>0\). Let \(K=W(k)[1/p]\). Choose its algebraic closure \(\bar{K}\) and set \(\Gamma_K(\bar{K}/K)\). For \(a \in \mathbb{Z}_{\geqslant 0}\), let \(\mathrm{M}\Gamma_{\mathbb{Q}_p}^{\mathrm{cr}}(a)\) be the category of crystalline \(\mathbb{Q}_p[\Gamma_K]\)-modules with Hodge-Tate weights from \([0,a]\). Define the full subcategory \(\mathrm{M}\Gamma_N^{\mathrm{cr}}(a)\) of the category of \(\Gamma_K\)-modules consisting of \(H=H_1/H_2\), where \(H_1\), \(H_2\) are \(\Gamma_K\)-invariant lattices in \(V \in \mathrm{M}\Gamma_N^{\mathrm{cr}}(a)\) and \(p^N H_1 \subset H_2 \subset H_1\).
J.-M. Fontaine conjectured in [Invent. Math. 81, 515–538 (1985; Zbl 0612.14043)] that the ramification subgroups \(\Gamma_K^{(v)}\) acts on \(H \in \mathrm{M}\Gamma_N^{\mathrm{cr}}(a)\) trivially if \(v > N - 1 + 1/(p-1)\). The author suggested in [Invent. Math. 101, No. 3, 631–640 (1990; Zbl 0761.14006)] a proof of this conjecture under the assumption \(0 \leqq a \leqq p-2\). The proof has a gap and is valid if we replace \(\Gamma_K^{(v)}\) by \(\Gamma_K^{(v)} \cap \Gamma_{K(\zeta_{N+1})}\), where \(\zeta_{N+1}\) is a primitive \(p^{N+1}\)-th root of unity. In the present paper, the author gives a new proof by using a non-canonical isomorphism between two complete discrete valuation fields of equal characteristic.
For a smooth proper scheme \(X_K\) over \(W(k)\), the above result gives the ramification estimates for the Galois equivalent subquotients of the étale cohomology \(H^a(X_{\bar{K}},\mathbb{Q}_p)\).
We should remark that the numeration of ramification subgroups \(\Gamma_K^{(v)}\) in [Zbl 0612.14043] is equal to the standard numeration [J.-P. Serre, Local fields. Translated from the French by Marvin Jay Greenberg. New York, Heidelberg, Berlin: Springer-Verlag (1979; Zbl 0423.12016)] plus one. Hence our ramification estimates should be plus one for a comparison.

Reviewer: Manabu Yoshida (Kumamoto)

MSC:

11S15	Ramification and extension theory
11S20	Galois theory

Keywords:

ramification; local fields; crystalline representation

Citations:

Zbl 0612.14043; Zbl 0761.14006; Zbl 0423.12016

References:

[1]	Abrashkin, V., Ramification in etale cohomology, Invent. Math., 101, 631-640, (1990) · Zbl 0761.14006
[2]	Abrashkin, V., Ramification filtration of the Galois group of a local field. III, Izv. Ross. Akad. Nauk Ser. Mat., Izv. Math., 62, 5, 857-900, (1998), English transl.: · Zbl 0918.11060
[3]	Caruso, X.; Liu, T., Some bounds for ramification of p-torsion semi-stable representations, J. Algebra, 325, 70-96, (2011) · Zbl 1269.14001
[4]	Deligne, P., LES corps locaux de caractéristique p, limites de corps locaux de caractéristique 0, (Representations of Reductive Groups Over a Local Field, Travaux en Cours, (1973), Hermann Paris), 119-157
[5]	Fontaine, J.-M., Il n’y a pas de variété abelienne sur \(\mathbb{Z}\), Invent. Math., 101, 631-640, (1990)
[6]	Fontaine, J.-M.; Laffaille, G., Construction de représentations p-adiques, Ann. Sci. Éc. Norm. Supér. (4), 15, 547-608, (1982) · Zbl 0579.14037
[7]	Hattori, Sh., On a ramification bound of torsion semi-stable representations over a local field, J. Number Theory, 129, 10, 2474-2503, (2009) · Zbl 1205.11127
[8]	Serre, J.-P., Local fields, (1980), Springer-Verlag Berlin, New York
[9]	Wintenberger, J.-P., Le corps des normes de certaines extensions infinies des corps locaux; application, Ann. Sci. Éc. Norm. Supér. (4), 16, 59-89, (1983) · Zbl 0516.12015
[10]	Wintenberger, J.-P., Extensions de Lie et groupes d’automorphismes des corps locaux de caractéristique p, C. R. Acad. Sci. Paris Sér. A-B, 288, 9, A477-A479, (1979), (in French)