Semi-Continuity for Conductor Divisors of Étale Sheaves

Abstract In this article, we prove a semi-continuity property for both conductor divisors and logarithmic conductor divisors for étale sheaves on higher relative dimensions in a geometric situation. It generalizes a semi-continuity result for conductors of étale sheaves on relative curves to higher relative dimensions, and it can be considered as a higher dimensional $\ell $-adic analogy of André’s result on the semi-continuity of Poincaré–Katz ranks of meromorphic connections on smooth relative curves.