Homological algebra in the category \(\text{Mod}_{(R\#U(g))}\) (Q1375370)

Let \(L\) be a finite dimensional Lie algebra over the algebraically closed field \(k\) of characteristic zero, and let \(R\) be a Noetherian \(k\)-algebra on which \(L\) acts by derivations. The author studies the category \(\mathcal C\) of \(R\)-modules which are compatible \(L\)-modules, which as \(L\)-modules are directed unions of finite dimensional \(L\)-modules. If \(U(L)\) is the universal enveloping algebra of \(L\) over \(k\), compatible modules are the same as modules over the smash product algebra \(R\#U(L)\). Let \((\cdot)^L\) denote taking \(L\)-invariants and let \(S=R^L\). Assume that \(S\) is also Noetherian. Then \(M\mapsto M^L\) defines a functor from \(R\#U(L)\)-modules to \(S\#U(L)\)-modules. When \(R\) is a directed union of finite dimensional \(L\)-modules, then \(R\otimes_S(\cdot)\) defines a functor from \(S\)-modules to the category \(\mathcal C\), and the author also defines an adjoint functor \({\mathcal F}_S(R,\cdot)\). The author establishes that \({\mathcal F}_S(R,\cdot)\) carries injectives to injectives and essential monomorphisms to essential monomorphisms. He further shows that it carries injective envelopes to injective envelopes. He shows that the functor \((\cdot)^L\) carries the injectives which are images by \({\mathcal F}_S(R,\cdot)\) of their submodules of invariants to injective \(S\)-modules. Then he considers the case that \(R\) is commutative. In this case \((\cdot)^L\) carries (some) projectives to projectives which is used to analyze \(\text{Pic}(S)\). The author further restricts to the case that \(R\) is the continuous dual of \(U(L)\) and obtains descriptions of projectives. Finally, he considers the case where \(L\) is semi-simple and for all finite type \(M\) \(\Hom_R(M,\cdot)\) is exact to \(L\)-modules, and shows that the category \(\mathcal C\) is semi-simple.