A note on Lie algebra cohomology (Q2027518)

Let \(L\) be a finite dimensional Lie algebra over a field \(k\) with universal enveloping algebra \(U(L)\). Denote the augmentation ideal of \(U(L)\) by \(I\). A (left) \(U(L)\)-module \(M\) is said to be \(I\)-torsion if for each \(m \in M\), there is \(n > 0\) with \(I^nm = 0\). Consider the category \(U(L)\)-mod of finitely generated \(U(L)\)-modules and the Serre subcategory (\(U(L)\)-mod)\(_I\) of \(I\)-torsion modules. The fundamental question of interest is whether the Ext-groups \(\operatorname{Ext}^j(k,k)\) (for the trivial module \(k\)) are the same in these two categories. The authors approach the problem by considering a functor on bounded derived categories: \[ \Phi_{L} : D^b((U(L)\text{-mod})_I) \to D_I^b(U(L)\text{-mod}), \] where the latter category is the full triangulated subcategory consisting of complexes with \(I\)-torsion cohomology. The question is to determine when \(\Phi_L\) is an equivalence. The authors observe that this holds if the graded Rees algebra \(\displaystyle U(L)^* := \bigoplus_{n \geq 0}I^n\) is graded left Noetherian. This latter condition is known to hold if \(L\) is nilpotent [\textit{J. T. Stafford} and \textit{N. R Wallach}, Trans. Am. Math. Soc. 272, 333--350 (1982; Zbl 0493.17004)], and the authors show the converse, that it is Noetherian only if the Lie algebra is nilpotent. More significantly, they show that \(\Phi_L\) is a categorical equivalence for other (i.e., non-nilpotent) Lie algebras, determining precise conditions on such \(L\). Set \(L_1 = L\) and, for \(n > 1\), \(L_n = [L, L_{n-1}]\). Further, set \(L_{\infty} := \cap_nL_n\) and \(L_{\text{nil}} := L/L_{\infty}\). The quotient map \(L \to L_{\text{nil}}\) induces a map in cohomology \(H^*(L_{\text{nil}},k) \to H^*(L,k)\), and the authors show that \(\Phi_L\) is an equivalence if and only if this latter map is an isomorphism. It is further shown that this is equivalent to \(H^{> 0}(L_{\infty},k)^{L_{\text{nil}}}\) being zero. The authors provide some small examples and related observations. For example, it is observed that in characteristic zero \(\Phi_L\) being an equivalence implies \(L\) is solvable. \par Lastly, assume that \(k\) is algebraically closed of characteristic zero, and let \(V\) be a unipotent linear algebraic group over \(k\). As an application of the main result, the authors show that the cohomology of a \(V\)-module may be identified with a cohomology group over the Lie algebra \(\operatorname{Lie} V\) of \(V\). This allows one to translate the computation of the cohomology of a quasicoherent equivariant sheaf over \(V\) to a cohomology computation over \(\operatorname{Lie} V\).