Equivariant Cohomology over Lie Groupoids and Lie-Rinehart Algebras

摘要

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 Omega and a smooth Omega-manifold f : M -> B-Omega over the space B-Omega of objects of Omega, the resulting Omega-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 Omega of objects of Omega; 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.