  We consider the extension problem for Lie algebroids over schemes over a
field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of
finitely generated Lie O-algebras L, we determine the obstruction to the
existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions
in terms of a suitable Lie algebroid hypercohomology group. In the preliminary
sections we study free Lie algebroids and recall some basic facts about Lie
algebroid hypercohomology.
