Ext-groups for modules

Definition 8.5.1 (Ext-groups for modules). Let \(R\) be an associative ring spectrum and let \(M, N \in \LMod _R\) be left \(R\)-modules. We define the \(n\)-th Ext-group of \(M\) and \(N\) over \(R\) to be \[ \Ext ^n_R(M,N) \quad := \quad \Ext ^n_{\LMod _R}(M,N) \quad = \quad \pi _{-n}\hom _R(M,N) \quad \simeq \quad \pi _0 \Hom _{\LMod _R}(M,N[n]). \] These are the Ext-groups from Definition 6.4.6 specialized to the stable \(\infty \)-category \(\LMod _R\).