Long exact sequence of Tor-groups

Observation 6.5.7 (Tor long exact sequence). Every exact sequence \(N' \to N \to N''\) in \(D\) induces an exact sequence \(M \otimes N' \to M \otimes N \to M \otimes N''\) in \(E\) (since \(M \otimes (-)\) is exact), and thus a long exact sequence of homotopy groups of the form \[ \begin {aligned} \dots \to \Tor ^{\otimes }_{n+1}(M,N'') \to \Tor ^{\otimes }_{n}(M,N') &\to \Tor ^{\otimes }_{n}(M,N) \\ &\to \Tor ^{\otimes }_{n}(M,N'') \to \Tor ^{\otimes }_{n-1}(M,N') \to \dots . \end {aligned} \] As a special case, given an object \(N \in D^{\heartsuit }\) and a short exact sequence \(M' \hookrightarrow M \twoheadrightarrow M''\) in \(C^{\heartsuit }\), the negative Tor-groups vanish, giving a one-sided long exact sequence of the form \[ \begin {aligned} \dots \to \Tor ^{\otimes }_1(M,N) \to \Tor ^{\otimes }_1(M'',N) &\to \Tor ^{\otimes }_{0}(M',N) \\ &\to \Tor ^{\otimes }_{0}(M,N) \to \Tor ^{\otimes }_{0}(M'',N) \to 0. \end {aligned} \] The same remarks apply to the other variable.