Long exact sequence of Ext-groups

Observation 6.4.10 (Ext long exact sequence). Every exact sequence \(Y' \to Y \to Y''\) in \(C\) induces an exact sequence of mapping spectra \(\hom _C(X,Y') \to \hom _C(X,Y) \to \hom _C(X,Y'')\), thus giving rise to a long exact sequence of homotopy groups of the form

Similarly, every exact sequence \(X' \to X \to X''\) in \(C\) induces a long exact sequence of the form \[ \begin {aligned} \dots \to \Ext ^{n-1}_C(X',Y) \to \Ext ^n_C(X'',Y) \to \Ext ^n_C(X,Y) \to \Ext ^n_C(X',Y) \to \Ext ^{n+1}_C(X'',Y) \to \dots . \end {aligned} \] As a special case, given an object \(N \in C^{\heartsuit }\) and a short exact sequence \(M' \hookrightarrow P \twoheadrightarrow M\) in \(C^{\heartsuit }\), the negative Ext-groups vanish, giving a one-sided long exact sequence of the form \[ \begin {aligned} 0 \to \Hom _{C^{\heartsuit }}(&M,N) \to \Hom _{C^{\heartsuit }}(P,N) \to \Hom _{C^{\heartsuit }}(M',N) \\ &\to \Ext ^1_C(M,N) \to \Ext ^1_C(P,N) \to \Ext ^1_C(M',N) \\ &\to \Ext ^2_C(M,N) \to \dots . \end {aligned} \]