Does the category of representations over an augmented algebra over a commutative ring have enough projectives?
I think it would be easier to begin with my motivation. Let be a commutative ring and a Lie algebra over . In Weibel's Homological algebra he states that the category of -representations (which he calls g-modules; they are -modules with a action compatible with the of ) has enough projectives and injectives. To prove this, he shows that the category of representations over g and the category of representations over are naturally isomorphic, where Ug is the universal enveloping algebra.
He claims this is enough, but I don't understand why. I turned to Cartan & Eilenberg, where they make the argument that, because is an augmented -algebra, we can compute for example for any a left module by using a projective resolution of as a module. This is only useful for defining homology as such. But does it show that the category of -representations has enough projective objects (which is required for Weibel's definition, since he uses another derived functor)?