A note on lattices in semi-stable representations. (English) Zbl 1208.14017
Let \(K\) be a complete discrete valuation field of characteristic \(0\) with a perfect residue field \(k\) of characteristic \(p>0\), \(\overline{K}\) be an algebraic closure of \(K\), and \(G_K=\mathrm{Gal}(\overline{K}/K)\). Since Fontaine introduced the notion of semi-stable \(p\)-adic representations of \(G_K\), many authors (for instance, Fontaine and Laffaille, Wach, Breuil, Berger) have tried to give a classification of \(G_K\)-stable \({\mathbb Z}_p\)-lattices in such \(p\)-adic Galois representations. However, all these results have certain restrictions on the absolute ramification index of \(K\) or on the Hodge-Tate weights. In the paper under review, the author gives such a classification without any restrictions.
The author’s construction is based on the theory of Kisin-modules. Let \(\pi\) be a uniformizer of \(K\), \(E(u)\) be its Eisenstein polynomial over \(K_0=W(k)[1/p]\), \(K_{\infty}=\bigcup_{n\geq 1}K({\pi}^{p^{-n}})\), \(G_{\infty}=\mathrm{Gal}(\overline{K}/K_{\infty})\), and \(\mathfrak{S}=W(k)[[u]]\). Let \(\varphi\) be the endomorphism of \(\mathfrak{S}\) which acts on \(W(k)\) via Frobenius and sends \(u\) to \(u^p\). For each integer \(r\geq 1\), a (finite free) Kisin module of height \(r\) is a finite free \(\mathfrak{S}\)-module \(\mathfrak{M}\) equipped with a \(\varphi\)-linear endomorphism \(\varphi_{\mathfrak{M}}:\mathfrak{M}\rightarrow \mathfrak{M}\) such that the submodule generated by the image of \(\varphi_{\mathfrak{M}}\) contains \(E(u)^r\mathfrak{M}\). To each finite free Kisin module of height \(r\), we can associate a finite free \({\mathbb Z}_p\)-representation of \(G_{\infty}\). In his paper [Crystalline representations and \(F\)-crystals. Algebraic geometry and number theory. Prog. Math. 253, 457–496 (2006; Zbl 1184.11052)], M. Kisin proved that any \(G_{\infty}\)-stable \({\mathbb Z}_p\)-lattice in a semi-stable \(p\)-adic Galois representation with Hodge weights in \(\{0,1,\cdots,r\}\) comes from a Kisin-module of height \(r\). To classify \(G_K\)-stable \({\mathbb Z}_p\)-lattices, the author considers the field \(\hat{K}=\bigcup_{n\geq 1}K({\pi}^{p^{-n}},\zeta_{p^n})\), where \(\zeta_{p^n}\) is a primitive \(p^n\)-th root of unity, and \(\hat{G}=\mathrm{Gal}(\hat{K}/K)\). Note that \(\hat{K}/K\) is the Galois closure of \(K_{\infty}/K\), and the author showed in another paper [Compos. Math. 144, No. 1, 61–88 (2008; Zbl 1133.14020)] that \(\hat{G}\simeq {\mathbb Z}_p(1)\rtimes H_K\) if \(p\geq 3\), where \(H_K=\mathrm{Gal}(\hat{K}/K_{\infty})\) is naturally isomorphic to the Galois group of the cyclotomic tower \(K(\mu_{p^{\infty}})=\bigcup_{n\geq 1}K(\zeta_{p^n})\) over \(K\). The author then introduces a (non-noetherian) ring \(\hat{\mathcal{R}}\) containing \(\mathfrak{S}\), equipped with a continuous action of \(\hat{G}\) and an endomorphism \(\varphi:\hat{\mathcal{R}}\rightarrow \hat{\mathcal{R}}\), called the Frobenius on \(\hat{\mathcal{R}}\), which extends the \(\varphi\) on \(\mathfrak{S}\) and commutes with the action of \(\hat{G}\). He defines a \((\varphi,\hat{G})\)-module of height \(r\) to be a finite free Kisin module \((\mathfrak{M},\varphi_{\mathfrak{M}})\) of height \(r\) together with a semi-linear action of \(\hat{G}\) on \(\hat{\mathcal{R}}\otimes_{\varphi,\mathfrak{S}}\mathfrak{M}\) compatible with Frobenius, such that the induced action of \(H_K\) on \(\mathfrak{M}\) and that of \(\hat{G}\) on \(\mathfrak{M}/u\mathfrak{M}\) are both trivial. The main result of this paper is that the category of \(G_K\)-stable \({\mathbb Z}_p\)-lattices in semi-stable representations with Hodge-Tate weights in \(\{0, 1, \cdots, r\}\) is anti-equivalent to the category of \((\varphi,\hat{G})\)-modules of height \(r\). The proof of this result uses a lot of ideas that appeared already in an earlier paper by the author [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 4, 633–674 (2007; Zbl 1163.11043)], and the proof for the case \(p=2\) is slightly more complicated due to the failure of the isomorphism \(\hat{G}\simeq \mathbb Z_p(1)\rtimes H_K\) in this case. In the last section of this paper, the author remedies a gap in loc. cit. related to the triviality condition on the action of \(\hat{G}\) on \(\mathfrak{M}/u\mathfrak{M}\) in the definition of a \((\varphi, \hat{G})\)-module.
The author’s construction is based on the theory of Kisin-modules. Let \(\pi\) be a uniformizer of \(K\), \(E(u)\) be its Eisenstein polynomial over \(K_0=W(k)[1/p]\), \(K_{\infty}=\bigcup_{n\geq 1}K({\pi}^{p^{-n}})\), \(G_{\infty}=\mathrm{Gal}(\overline{K}/K_{\infty})\), and \(\mathfrak{S}=W(k)[[u]]\). Let \(\varphi\) be the endomorphism of \(\mathfrak{S}\) which acts on \(W(k)\) via Frobenius and sends \(u\) to \(u^p\). For each integer \(r\geq 1\), a (finite free) Kisin module of height \(r\) is a finite free \(\mathfrak{S}\)-module \(\mathfrak{M}\) equipped with a \(\varphi\)-linear endomorphism \(\varphi_{\mathfrak{M}}:\mathfrak{M}\rightarrow \mathfrak{M}\) such that the submodule generated by the image of \(\varphi_{\mathfrak{M}}\) contains \(E(u)^r\mathfrak{M}\). To each finite free Kisin module of height \(r\), we can associate a finite free \({\mathbb Z}_p\)-representation of \(G_{\infty}\). In his paper [Crystalline representations and \(F\)-crystals. Algebraic geometry and number theory. Prog. Math. 253, 457–496 (2006; Zbl 1184.11052)], M. Kisin proved that any \(G_{\infty}\)-stable \({\mathbb Z}_p\)-lattice in a semi-stable \(p\)-adic Galois representation with Hodge weights in \(\{0,1,\cdots,r\}\) comes from a Kisin-module of height \(r\). To classify \(G_K\)-stable \({\mathbb Z}_p\)-lattices, the author considers the field \(\hat{K}=\bigcup_{n\geq 1}K({\pi}^{p^{-n}},\zeta_{p^n})\), where \(\zeta_{p^n}\) is a primitive \(p^n\)-th root of unity, and \(\hat{G}=\mathrm{Gal}(\hat{K}/K)\). Note that \(\hat{K}/K\) is the Galois closure of \(K_{\infty}/K\), and the author showed in another paper [Compos. Math. 144, No. 1, 61–88 (2008; Zbl 1133.14020)] that \(\hat{G}\simeq {\mathbb Z}_p(1)\rtimes H_K\) if \(p\geq 3\), where \(H_K=\mathrm{Gal}(\hat{K}/K_{\infty})\) is naturally isomorphic to the Galois group of the cyclotomic tower \(K(\mu_{p^{\infty}})=\bigcup_{n\geq 1}K(\zeta_{p^n})\) over \(K\). The author then introduces a (non-noetherian) ring \(\hat{\mathcal{R}}\) containing \(\mathfrak{S}\), equipped with a continuous action of \(\hat{G}\) and an endomorphism \(\varphi:\hat{\mathcal{R}}\rightarrow \hat{\mathcal{R}}\), called the Frobenius on \(\hat{\mathcal{R}}\), which extends the \(\varphi\) on \(\mathfrak{S}\) and commutes with the action of \(\hat{G}\). He defines a \((\varphi,\hat{G})\)-module of height \(r\) to be a finite free Kisin module \((\mathfrak{M},\varphi_{\mathfrak{M}})\) of height \(r\) together with a semi-linear action of \(\hat{G}\) on \(\hat{\mathcal{R}}\otimes_{\varphi,\mathfrak{S}}\mathfrak{M}\) compatible with Frobenius, such that the induced action of \(H_K\) on \(\mathfrak{M}\) and that of \(\hat{G}\) on \(\mathfrak{M}/u\mathfrak{M}\) are both trivial. The main result of this paper is that the category of \(G_K\)-stable \({\mathbb Z}_p\)-lattices in semi-stable representations with Hodge-Tate weights in \(\{0, 1, \cdots, r\}\) is anti-equivalent to the category of \((\varphi,\hat{G})\)-modules of height \(r\). The proof of this result uses a lot of ideas that appeared already in an earlier paper by the author [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 4, 633–674 (2007; Zbl 1163.11043)], and the proof for the case \(p=2\) is slightly more complicated due to the failure of the isomorphism \(\hat{G}\simeq \mathbb Z_p(1)\rtimes H_K\) in this case. In the last section of this paper, the author remedies a gap in loc. cit. related to the triviality condition on the action of \(\hat{G}\) on \(\mathfrak{M}/u\mathfrak{M}\) in the definition of a \((\varphi, \hat{G})\)-module.
Reviewer: Yichao Tian (Princeton)
MSC:
| 14F30 | \(p\)-adic cohomology, crystalline cohomology |
| 14L05 | Formal groups, \(p\)-divisible groups |
| 11S23 | Integral representations |
| 11S20 | Galois theory |
References:
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