A classical theorem of Lefschetz asserts that non-trivial line bundles on a smooth projective variety of dimension >= 3 remain non-trivial upon restriction to an ample divisor. In SGA2, Grothendieck recast this result in purely local terms. Answering a question raised recently by Kollar, we will explain how this local reformulation remains true under much milder hypotheses than those in SGA2. Our method uses a vanishing theorem in characteristic p, and formal geometry over certain very large (non-noetherian) schemes. This is joint work with Johan de Jong.