The Serre spectral sequence of a Lie subalgebroid

Abstract

We study a spectral sequence approximating Lie algebroid cohomology associated to a Lie subalgebroid. This is a simultaneous generalisation of several classical constructions in differential geometry, including the Leray-Serre spectral sequence for de Rham cohomology associated to a fibration, the Hochschild-Serre spectral sequence for Lie algebras, and the Mackenzie spectral sequence for Lie algebroid extensions. We show that, for wide Lie subalgebroids, the spectral sequence converges to the Lie algebroid cohomology, and that, for Lie subalgebroids over proper submanifolds, the spectral sequence converges to the formal Lie algebroid cohomology. We discuss applications and recover several constructions in Poisson geometry in which this spectral sequence has appeared naturally in the literature.