##
**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].

– 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) |