Injective dimension of crossed products (Q1911750)

The article under review investigates the injective dimension \(\text{id }S=\text{inj.dim}(_SS)\) of crossed products \(S=R\#_\sigma U\), where \(U=U({\mathfrak g})\) is the enveloping algebra of a finite dimensional Lie \(k\) algebra \(\mathfrak g\) and \(R\) is a commutative Noetherian \(k\)-algebra with \(\text{id }R<\infty\). The principal result are the inequalities \(\text{id }R\leq\text{id }S\leq\text{id }R+\dim_k{\mathfrak g}\). Reviewer's remark: The second inequality does in fact hold more generally for crossed products \(S=R\#_\sigma H\), where \(R\) is any left Noetherian \(k\)-algebra and \(H\) any Hopf \(k\)-algebra. Indeed, Proposition 2.3(a) of [\textit{M. E. Lorenz} and \textit{M. Lorenz}, Proc. Am. Math. Soc. 123, No. 1, 33-38 (1995; Zbl 0826.16037)] implies that \(\text{id }S\leq\text{id }R+\text{proj.dim }k_H\).