injective modules over a Lie groupoid

injective modules over a Lie groupoid

[Johannes Huebschmann]

Dear All

For a Lie group G and a vector space V,
C^{\infty}(G,V) is a differentiably injective G-module
(Hochschild-Mostow).

Is there an analoguous construction
for a Lie groupoid or, in the algebraic setting,
cogroupoid object in the category of commutative algebras?

Let G be a Lie groupoid, with object manifold G_o,
source and target maps being supposed surjective submersions.
A G-module is a vector bundle

V \to G_o

on G_o with a G-structure
(pairing G x_G_o V to V over G_o satisfying the obvious compatiblity
conditions).

If we start with a vector bundle V to G_o on G_o,
what corresponds to the construction

C^{\infty}(G,V)

for the special case where G is an ordinary Lie group?

More generally, G being a Lie groupoid,
does the category of G-modules have enough injectives?

Where in the literature can I find answers to these questions
if any?

Many thanks in advance

Regards

Johannes


HUEBSCHMANN Johannes
Professeur de Mathematiques
USTL, UFR de Mathematiques
UMR 8524 Laboratoire Paul Painleve
F-59 655 Villeneuve d'Ascq Cedex  France
http://math.univ-lille1.fr/~huebschm

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