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**Nearby cycles for log smooth families.**
*(English)*
Zbl 0926.14006

From the introduction: In this paper we calculate \(l\)-adic nearby cycles for log smooth families using log étale cohomology. The point is that, though our concerned families may not be smooth, they start to behave as if they were smooth once equipped with natural log structures. Then our calculation is as easy and transparent as that for usual smooth families. Main result:

Theorem. Let \(X\to S=\text{Spec}(A)\) be a morphism of schemes with \(A\) being a henselian discrete valuation ring. Let \(s\) be the closed point of \(S\) and \(\eta=\text{Spec} K\) the generic one. Let \(n\geq 1\) be an integer invertible on \(S\). For any \(q\in\mathbb{Z}\), we have the nearby cycle \(R^q\Psi_\eta \mathbb{Z}/n\mathbb{Z}\) on the product topos \(X_s\times_sS\) [cf. P. Deligne, in: Sém. Géom. algébrique 1967-1969, SGA 7 II, Lect. Notes Math. 340, Exposé XIII, 82-115 (1973; Zbl 0266.14008); 2.1.1]. Suppose that \(X\) has log smooth reduction in the sense explained below. Then the action on \(R^q\Psi_\eta \mathbb{Z}/n \mathbb{Z}\) of the wild inertia group \(P\) of \(K\) is trivial .

Corollary. We assume further that \(X\to S\) is proper. Then the action of \(P\) on \(H^q (X_{\overline\eta}, \mathbb{Z}/n \mathbb{Z})\) is trivial, where \(\overline\eta\) is the spectrum of a separable closure of \(K\).

We explain what is log smooth reduction. This means that étale locally on \(X\), \(X\) is étale over \(\text{Spec} (A[P]/(\pi-x))\). Here \(P\) is a finitely generated, saturated (commutative) monoid, \(\pi\) is a prime element of \(A\), and \(x\) is an element of \(P\) such that

(i) the order of the torsion part of \(P^{gp}/ \langle x\rangle\) is invertible on \(X\); and

(ii) for any \(a\in P\), there is an \(m\geq 1\) and \(b\in P\) such that \(ab=x^m\) in \(P\).

Theorem. Let \(X\to S=\text{Spec}(A)\) be a morphism of schemes with \(A\) being a henselian discrete valuation ring. Let \(s\) be the closed point of \(S\) and \(\eta=\text{Spec} K\) the generic one. Let \(n\geq 1\) be an integer invertible on \(S\). For any \(q\in\mathbb{Z}\), we have the nearby cycle \(R^q\Psi_\eta \mathbb{Z}/n\mathbb{Z}\) on the product topos \(X_s\times_sS\) [cf. P. Deligne, in: Sém. Géom. algébrique 1967-1969, SGA 7 II, Lect. Notes Math. 340, Exposé XIII, 82-115 (1973; Zbl 0266.14008); 2.1.1]. Suppose that \(X\) has log smooth reduction in the sense explained below. Then the action on \(R^q\Psi_\eta \mathbb{Z}/n \mathbb{Z}\) of the wild inertia group \(P\) of \(K\) is trivial .

Corollary. We assume further that \(X\to S\) is proper. Then the action of \(P\) on \(H^q (X_{\overline\eta}, \mathbb{Z}/n \mathbb{Z})\) is trivial, where \(\overline\eta\) is the spectrum of a separable closure of \(K\).

We explain what is log smooth reduction. This means that étale locally on \(X\), \(X\) is étale over \(\text{Spec} (A[P]/(\pi-x))\). Here \(P\) is a finitely generated, saturated (commutative) monoid, \(\pi\) is a prime element of \(A\), and \(x\) is an element of \(P\) such that

(i) the order of the torsion part of \(P^{gp}/ \langle x\rangle\) is invertible on \(X\); and

(ii) for any \(a\in P\), there is an \(m\geq 1\) and \(b\in P\) such that \(ab=x^m\) in \(P\).

### MSC:

14F20 | Étale and other Grothendieck topologies and (co)homologies |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14G20 | Local ground fields in algebraic geometry |

14C25 | Algebraic cycles |