Variation of the Swan conductor of an \(\mathbb{F}_{\ell}\)-sheaf on a rigid disc

Abstract

This article studies the variation of the Swan conductor of a lisse \'etale sheaf of \(\mathbb{F}_{\ell}\)-modules \(\mathcal{F}\) on the rigid unit disc \(D\) over a complete discrete valuation field \(K\) with algebraically closed residue field of characteristic \(p\neq \ell\). We associate to \(\mathcal{F}\) a function \({\rm sw}_{\rm AS}(\mathcal{F}, \cdot): \mathbb{Q}_{\geq 0}\to \mathbb{Q}\), defined with the Abbes-Saito logarithmic ramification filtration, which measures, at each \(t\in \mathbb{Q}_{\geq 0}\), the ramification of the restriction of \(\mathcal{F}\) to the subdisc of radius \(t\) along the special fiber of the normalized integral model. We prove that this function is continuous and piecewise linear, with finitely many slopes which are all integers. We compute the slope at \(t\in \mathbb{Q}_{\geq 0}\) in terms of a characteristic cycle associated to \(\mathcal{F}\), a (power of a) logarithmic differential form defined by ramification theory.