## 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.

### 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
Full Text:

### References:

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