The classical Dade’s Lemma says that over group algebras of elementary abelian \(p\)-groups, a finitely generated module is projective if and only if its restriction to all the cyclic shifted subgroups are projective, provided that the ground field is algebraically closed. For local complete intersections (that is, quotients of regular local rings by ideals generated by regular sequences), a cohomological generalization of Dade’s Lemma is proved implicitly by L. L. Avramov [Invent. Math. 96, No. 1, 71–101 (1989; Zbl 0677.13004)], and by L. L. Avramov and R.-O. Buchweitz [Invent. Math. 142, No. 2, 285–318 (2000; Zbl 0999.13008)]. In the paper under review the authors further generalize Dade’s Lemma to quotients of arbitrary local rings, and provide both a homological and a cohomological version. More precisely, let \((S, n, k)\) be a local ring with \(k\) algebraically closed, \(I\subseteq S\) an ideal generated by a regular sequence, and denote by \(R\) the quotient \(S/I\). The authors provide criteria for the vanishing of \(\mathrm{Tor}^{R}_{n}(M,N)\) and \(\mathrm{Ext}_{R}^{n}(M,N)\) for \(n\gg 0\), where \(M\) and \(N\) are two finitely generated \(R\)-modules, in terms of the vanishing of \(\mathrm{Tor}^{S/(f)}_{n}(M,N)\) and \(\mathrm{Ext}_{S/(f)}^{n}(M,N)\) for \(n\gg 0\) and all \(f\in I\backslash nI\).