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Crystalline representations of torsion. (Représentations cristallines de torsion.) (French) Zbl 0902.11051
Let $$p$$ be a prime number, and let $$K_0$$ be a field of characteristic zero, complete with respect to a discrete valuation, absolutely non-ramified with perfect residue field $$k$$ of characteristic $$p$$. Fix an algebraic closure $$\overline{K}$$ of $$K_0$$ and let $$G_{K_0}=\text{Gal}(\overline{K}/K_0)$$. One would like to describe $$p$$-adic representations of $$G_{K_0}$$, in particular, one may ask when these come from algebraic geometry, and if so, can they be described by simpler objects. Here the question for crystalline representations in terms of Fontaine’s $$(\varphi,\Gamma)$$-modules is answered.
For an integer $$n\geq 1$$, one writes $$K_n\subset\overline{K}$$ for the subfield generated over $$K_0$$ by the $$p^n$$-th roots of unity, ($$p\neq 2$$). Write $$K_{\infty}=\cup_{n\geq 1}K_n$$, $$\Gamma_{K_0}=\text{Gal}(\overline{K}/K_{\infty})$$ and $$\Gamma_f=\Gamma_{K_0,\text{tors}}$$. Define $$\Gamma_0=\Gamma/\Gamma_f$$. $$\Gamma_0$$ can be identified with the Galois group of the cyclotomic $${\mathbb{Z}}_p$$-extension of $$K_0$$ contained in $$K_{\infty}$$, $$p\neq 2$$. For $$p=2$$ one defines $$\Gamma_0=\Gamma$$. J.-M. Fontaine defined an action of $$\Gamma_0$$ on a certain ring $$S_0$$ [The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 249-309 (1991; Zbl 0743.11066)]. Then, for $$h\in{\mathbb{N}}$$, one may define the full subcategory $$\Gamma_0\Phi{\mathcal M} ^h_{S_0}$$ of the category of $$(\varphi,\Gamma_0)$$-modules over $$S_0$$, and one writes $$\Gamma_0\Phi{\mathcal M}^h_{S_0,\text{tors}}$$ for the full subcategory whose objects are of $$p$$-torsion.
Let $$\Gamma_0\Phi {\mathcal M}^+_{S_0}=\cup_{h\in{\mathbb{N}}}\Gamma_0\Phi{\mathcal M}^h_{S_0}$$ and define the cr-height of an object $$M$$ of $$\Gamma_0\Phi{\mathcal M}^+_{S_0}$$ as the smallest $$h$$ such that $$M$$ is an object of $$\Gamma_0\Phi{\mathcal M}^h_{S_0}$$. One constructs an exact and faithful functor $V^*_{\text{cr}}:\Gamma_0\Phi{\mathcal M}^h_{S_0,\text{tors}}\rightarrow \text{Rep}_{{\mathbb{Z}}_p,\text{tors}}(G_{K_0})$ whose essential image is denoted $$\text{Rep}^h_{{\mathbb{Z}}_p,\text{tors,cr}}(G_{K_0})$$, and is called the category of representations of finite cr-height $$\leq h$$.
The main result of the paper is: For an integer $$h$$ with $$0\leq h\leq p-1$$ and a representation $$T$$ in $$\text{Rep}_{{\mathbb{Z}}_p, \text{tors}}(G_{K_0})$$, one has that $$T$$ is in $$\text{Rep}^h_{{\mathbb{Z}}_p, \text{tors,cr}} (G_{K_0})$$ iff $$T$$ is isomorphic to a subquotient of a crystalline representation with Hodge-Tate weights in $$[0,h]$$. A $$p$$-adic representation $$V$$ of $$G_{K_0}$$ is said to be of cr-height $$\leq h$$ if it is “isomorphic” with some object $$M$$ of $$\Gamma_0\Phi{\mathcal M}^h_{S_0}$$ (this can be made precise). One also has the notion of finite height for a $$p$$-adic representation $$V$$ of $$G_{K_0}$$. Then, if $$V$$ is of finite cr-height, it is of finite height. One obtains: Let $$r,h\in{\mathbb{Z}}$$, $$0\leq h\leq p-1$$, and let $$V$$ be a $$p$$-adic representation of $$G_{K_0}$$. Then $$V$$ is crystalline with Hodge-Tate weights in $$[r,r+h]$$ iff $$V(r)$$ is a representation of finite cr-height $$\leq h$$. Moreover, $$V$$ is of finite height.
The proof of the main result is surprisingly long and technical. First, a detailed analysis of crystalline torsion representations is given to obtain the exact faithfulness of a functor $$V^*_{\text{cris}}$$ restricted to a category of filtered modules $${\mathcal {MF}}^h_{W,\text{tors}}$$ over the Witt ring $$W$$ of $$k$$, of height $$\leq h$$, $$h\leq p-1$$, with values in $$\text{Rep}_{{\mathbb{Z}}_p,\text{tors}} (G_{K_0})$$. For $$h\leq p-2$$, $$V^*_{\text{cris}}$$ is even fully faithful. If $$\Lambda$$ in $${\mathcal {MF}}^h_{\text{tors}}$$ has finite length, then $$\text{length}_W\Lambda=\text{length}_{{\mathbb{Z}}_p}V^*_{\text{cris}}(\Lambda)$$, and if $$\Lambda$$ is free over $$W$$, $$V^*_{\text{cris}}(\Lambda)$$ is free over $${\mathbb{Z}}_p$$ and $$\text{rank}_W(\Lambda)=\text{rank}_{{\mathbb{Z}}_p}V^*_{\text{cris}}(\Lambda)$$. As a matter of fact, a construction of the rings of $$p$$-adic periods $$A_{\text{cris}}$$ and $$B^+_{\text{cris}}$$ à la Fontaine-Laffaille is given in a slightly different way. The next step leads to the construction of a filtration on objects of $$\Gamma_0\Phi{\mathcal M}^h_{S_0}$$, $$h\leq p-1$$, giving rise to an additive functor $$i^*:\Gamma_0\Phi{\mathcal M}^h_{S_0}\rightarrow{\mathcal {MF}}^h_W$$, which is shown to be exact. It is faithful for $$h\leq p-2$$. For $$h=p-1$$ one must impose an additional condition. Actually, $$i^*$$ induces an equivalence between $$\Gamma_0\Phi{\mathcal M}^h_{S_0}$$ and $${\mathcal {MF}}^h_W$$. Finally, one must “identify” the two representations induced by $$V^*_{\text{cr}}$$ and $$V^*_{\text{cris}}$$.

##### MSC:
 11S23 Integral representations 14F30 $$p$$-adic cohomology, crystalline cohomology 14L05 Formal groups, $$p$$-divisible groups
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