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

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.


11S15 Ramification and extension theory
11S20 Galois theory
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