Lie algebroid cohomology and Lie algebroid extensions

We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme X, and a coherent sheaf of Lie OX-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.