Characteristic \(p\) analogue of modules with finite crystalline height
Let \(K\) be a complete valuation field with perfect residue field \(k\) of characteristic \(p>0\) and let \(K_{\text{sep}}\) be a separable closure of \(K\). Set \(\Gamma_ K={\text{Gal}}(K_{\text{sep}}/K)\) and let \(\Gamma_ K^ {(v)}\) be the ramification group with upper ramification number \(v\). Let \(R\) denote the valuation ring of \(K\). When \(K\) is of characteristic \(0\), let \({\text{MG}}_ K^ h\) be the category of finite \({\mathbb Z}_ p[\Gamma _ K]\)-modules with crystalline height \(h\) with \(h\in{\mathbb N}\). \textit{J.-M. Fontaine} [Invent. Math. 81, 515--538 (1985; Zbl 0612.14043)] proved that if \(G\in {\text{MG}} _ K^ 1\), \(p^ N G= 0\) and \(v>e(N+1/(p-1)) -1\), then \(\Gamma _ K^ {(v)}\) acts trivially on \(G\). Here \(e\) denotes the ramification index of \(K\) over \({\mathbb Q}_ p\). When \(K\) is of characteristic \(p\), the author [Compos. Math. 142, No. 4, 867--888 (2006; Zbl 1102.14032)] proved that if \(G={\mathcal G}(K_{\text{sep}})\) where \({\mathcal G}\) is an \(O\)-strict module over \(R\) (concept introduced by \textit{G. Faltings} [Mosc. Math. J. 2, No. 2, 249--279 (2002; Zbl 1013.11079)]) and \(\pi^ N G=0\), then for \(v>e(N+1/(q-1))-1\), \(\Gamma _ K^{(v)}\) acts trivially on \(G\), where \(k\supseteq {\mathbb F}_ q\), \(O={\mathbb F}_ q\pi\subseteq R\) and \(\pi\in R\) is not invertible. This result is the analogue to Fontaine's estimate in the unequal characteristic case. In the equal characteristic case, to see the interpretation of higher étale cohomology, it seems natural to consider Galois modules arising from Anderson motives. Let \({\text{MG}}(O)_ K\) be the category of \(O\)-modules of finite type with continuous \(O\)-linear action of \(\Gamma _ K\). Then \({\text{MG}}^ h(O)_ K\) is the corresponding category of \(O[\Gamma _ K]\)-modules of ``finite crystalline height'' \(h\), with \(h\in {\mathbb N}\). The first main result in this paper is that if \(G\in {\text{MG}}^ h(O)_ K\) then there is \(\pi\)-divisible group \(\{G_ n, i_ n\}_{n\geq 1}\) in \({\text{MG}}^ h(O)_ K\) such that if \(N\in {\mathbb N}\) is such that \(\pi^ N {\text{id}} _ G=0\), then there is an embedding of \(G\) into \(G_ N\) in the category \({\text{MG}} (O)_ K\). The second main theorem establishes that if \(G\in {\text{MG}}^ h(O)_ K\) and \(\pi ^ N G=0\) then for every \(v>e(N-1+hq/(q-1))-1\), \(\Gamma_ K^{(v)}\) acts trivially on \(G\), where \(e\) is the ramification index of \(K\) over the field of quotients of \(O\).
scientific article; zbMATH DE number 5543156