Spreading out perverse sheaves

Abstract

We spread out geometrically irreducible perverse sheaves. Over an arithmetic base scheme, such sheaves extend from the generic fiber to a relatively perverse, universally locally acyclic sheaf over a dense open of the base scheme. As an application, we prove the arithmetic Kashiwara conjecture by Esnault and Kerz for geometric traits in equicharacteristic and generalize the decomposition theorem for arithmetic complexes to fields finitely generated over a separably closed field. We also recover the Hard Lefschetz theorem for arithmetic perverse sheaves.