Lie algebroid cohomology and Lie algebroid extensions

Abstract 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 O X -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.