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Exposé V: Semi-stable reduction and crystalline cohomology with logarithmic poles. (English) Zbl 0852.14004
Fontaine, Jean-Marc (ed.), Périodes $$p$$-adiques. Séminaire de Bures-sur-Yvette, France, 1988. Paris: Société Mathématique de France, Astérisque. 223, 221-268 (1994).
We say a scheme $$X$$ over a discrete valuation ring $$A$$ is with semi-stable reduction if étale locally on $$X$$, there is a smooth morphism $$X\to \text{Spec} (A [T_1, \dots, T_r ]/( T_1 \cdots T_r- \pi)$$ for some $$r\geq 0$$, where $$\pi$$ is a uniformizing parameter. This condition is equivalent to the condition that $$X$$ is regular, the generic fiber of $$X$$ is smooth, and the closed fiber of $$X$$ is a reduced divisor with normal crossings on $$X$$. Let $$A$$ be a complete discrete valuation ring with field of fractions $$K$$ and with residue field $$k$$ such that $$\text{char} (K) =0$$, $$\text{char} (k)= p>0$$, and $$k$$ is perfect, and let $$K_0$$ be the field of fractions of the ring $$W= W(k)$$ of Witt vectors. Let $$X$$ be a proper scheme over $$A$$ with semi-stable reduction, and let $$Y= X\otimes_A k$$. Then, the crystalline cohomology group $$H^m_{\text{crys}} (Y/W) \otimes_W K_0$$ $$(m\in \mathbb{Z})$$ is not a “good cohomology” when $$Y$$ is singular. However U. Jannsen conjectured [in: Galois groups over $$\mathbb{Q}$$, Proc. Workshop, Berkeley 1987, Publ., Math. Sci. Res. Inst. 16, 315-359 (1989; Zbl 0703.14010)] that there is a “new crystalline cohomology group” $$D$$, which is a finite-dimensional $$K_0$$-vector space endowed with
– a bijection frobenius-linear operator $$\varphi: D\to D$$ called the frobenius,
– a nilpotent operator $${\mathcal N}: D\to D$$ called the monodromy operator, satisfying $${\mathcal N} \varphi= p\varphi{\mathcal N}$$,
– a $$K$$-isomorphism with the de Rham cohomology $$\rho: D\otimes_{K_0} K\widetilde {\to} H^m_{DR} (X_K/ K)$$ $$(X_K= X\otimes_A K)$$.
This space $$D$$ is a mixed characteristic analogue of the limit Hodge structure. The triple $$(D, \varphi, {\mathcal N})$$ is constructed by O. Hyodo [Compos. Math. 78, No. 3, 241-260 (1991; Zbl 0742.14015)] by using some de Rham-Witt complex with logarithmic poles. In this paper, we give another construction of $$(D, \varphi, {\mathcal N})$$ using the crystalline cohomology theory with logarithmic poles and give the isomorphism $$\rho$$. The 4-ple $$(D, \varphi, {\mathcal N}, \rho)$$ has the following further properties:
– $$(D, \varphi, {\mathcal N})$$ depends only on the scheme $$X\otimes_A A/m^2_A$$ over $$A/m^2_A$$ where $$m_A$$ denotes the maximal ideal of $$A$$.
– The isomorphism $$\rho$$ depends on a choice of a prime element $$\pi$$ of $$A$$.
If we indicate the choice of $$\pi$$ as $$\rho_\pi$$, we have $$\rho_{\pi u}= \rho_\pi \circ \exp (\log (u){\mathcal N})$$ for $$u\in A^\times$$, where we denote the $$K$$-linear operator on $$D\otimes_{K_0} K$$ induced by $${\mathcal N}$$ by the same letter $${\mathcal N}$$. The $$K$$-linear operator $$\rho_\pi \circ {\mathcal N}\circ \rho_\pi^{-1}$$ on $$H^m_{DR} (X_K/K)$$ is independent of the choice of $$\pi$$. As is shown by O. Hyodo [loc. cit.], the triple $$(D, \varphi, {\mathcal N})$$ is $$\otimes_W K_0$$ of a triple $$(H, \varphi, {\mathcal N})$$ with $$H$$ a canonical defined $$W(k)$$-module of finite type. L. Illusie has proposed a method to show that the operator $${\mathcal N}: H\to H$$ is already nilpotent before $$\otimes_W K_0$$.
The theory of crystalline cohomology with logarithmic poles used in this paper is based on the theory of “logarithmic structures” of Fontaine-Illusie reported by K. Kato [in: Périodes $$p$$-adiques. Sém. Bures-sur-Yvette 1988, Astérisque 223, 269-293 (1994; Zbl 0847.14009)]. In fact, by using this theory of logarithmic structures, we construct $$(D, \varphi, {\mathcal N}, \rho)$$ in this paper not only for $$X$$ as above, but also for a scheme over $$A$$ with a “smooth logarithmic structure whose reduction is of Cartier type” (for example, a product of schemes with semi-stable reduction is such a scheme). We give also the detailed study of the de Rham-Witt complexes with logarithmic poles associated to such general situation.
The subject of this paper is studied independently by G. Faltings [in: The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 219-248 (1990; Zbl 0736.14004)].
For the entire collection see [Zbl 0802.00019].

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14L30 Group actions on varieties or schemes (quotients)