Equivariant Cohomology over Lie Groupoids and Lie–Rinehart Algebras

Abstract

Using the language and terminology of relative homological algebra, in particular that of derived functors, we introduce equivariant cohomology over a general Lie–Rinehart algebra and equivariant de Rham cohomology over a locally trivial Lie groupoid in terms of suitably defined monads (also known as triples) and the associated standard constructions. This extends a characterization of equivariant de Rham cohomology in terms of derived functors developed earlier for the special case where the Lie groupoid is an ordinary Lie group, viewed as a Lie groupoid with a single object; in that theory over a Lie group, the ordinary Bott–Dupont–Shulman–Stasheff complex arises as an a posteriori object. We prove that, given a locally trivial Lie groupoid Ω and a smooth Ω-manifold f : M → B Ω over the space B Ω of objects of Ω, the resulting Ω-equivariant de Rham theory of f reduces to the ordinary equivariant de Rham theory of a vertex manifold f −1(q) relative to the vertex group $$\Omega^q_q$$ , for any vertex q in the space B Ω of objects of Ω; this implies that the equivariant de Rham cohomology introduced here coincides with the stack de Rham cohomology of the associated transformation groupoid; thus this stack de Rham cohomology can be characterized as a relative derived functor. We introduce a notion of cone on a Lie–Rinehart algebra and in particular that of cone on a Lie algebroid. This cone is an indispensable tool for the description of the requisite monads.