connectingExtMap(M, g, f)
connectingExtMap(g, f, M)
Since Ext is a bifunctor, there are two different connecting maps. The first comes from applying the covariant functor $\operatorname{Hom}_S(M, -)$ to a short exact sequence of modules. The second comes from applying the contravariant functor $\operatorname{Hom}_S(-, M)$ to a short exact sequence of modules. More explicitly, given the short exact sequence
$\phantom{WWWW} 0 \leftarrow C \xleftarrow{g} B \xleftarrow{f} A \leftarrow 0, $
the $(-i)$-th connecting homomorphism is, in the first case, a map $\operatorname{Ext}_S^i(M, C) \to \operatorname{Ext}_S^{i+1}(M, A)$ and, in the second case, a map $\operatorname{Ext}_S^i(A, M) \to \operatorname{Ext}_S^{i+1}(C, M)$. Observe that the connecting homomorphism is indexed homologically, whereas Ext modules are indexed cohomologically, explaining the different signs for the index $i$.
As a first example, applying the functor $\operatorname{Hom}(S/I, -)$ to a short exact sequence of modules
$\phantom{WWWW} 0 \leftarrow S/h \leftarrow S \xleftarrow{h} S(- \deg h) \leftarrow 0 $
gives rise to the long exact sequence in Ext modules having the form
$\phantom{WWWW} \dotsb \leftarrow \operatorname{Ext}^{i+1}(S/I, S(-\deg h)) \xleftarrow{\delta_{-i}} \operatorname{Ext}^i(S/I, S/h) \leftarrow \operatorname{Ext}^i(S/I, S) \leftarrow \operatorname{Ext}^i(S/I, S(-\deg h)) \leftarrow \dotsb. $
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As a second example, applying the functor $\operatorname{Hom}(-, S)$ to the short exact sequence of modules
$\phantom{WWWW} 0 \leftarrow S/(I+J) \leftarrow S/I \oplus S/J \leftarrow S/I \cap J \leftarrow 0 $
gives rise to the long exact sequence of Ext modules having the form
$\phantom{WWWW} \dotsb \leftarrow \operatorname{Ext}^{i+1}(S/(I+J), S) \xleftarrow{\delta_{-i}} \operatorname{Ext}^i(S/I \cap J, S) \leftarrow \operatorname{Ext}^i(S/I \oplus S/J, S) \leftarrow \operatorname{Ext}^i(S/(I+J), S) \leftarrow \dotsb. $
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