Tor-groups for tensor products

Definition 6.5.6 (Tor-groups). In the situation of Definition 6.5.1, for objects \(M \in C\) and \(N \in D\), we define the \(n\)-th Tor-group to be \[ \Tor ^{\otimes }_n(M,N) \quad := \quad \pi _n(M \otimes N) \qin E^{\heartsuit }. \] When \(M \in C^{\heartsuit }\) and \(N \in D^{\heartsuit }\) both lie in the heart, we also define \(M \otimes ^{\heartsuit } N := \Tor ^{\otimes }_0(M,N) = \pi _0(M \otimes N)\), resulting in a functor \[ - \otimes ^{\heartsuit } - \colon C^{\heartsuit } \times D^{\heartsuit } \to E^{\heartsuit }. \] Since \(M \otimes N \in E_{\geq 0}\), this agrees with \(\tau _{\leq 0}(M \otimes N)\) and it follows that this functor is right exact in both variables.