Exact Sequences

Let \(R\) be a ring. A sequence \[ …​ \rightarrow M_{i-1} \rightarrow^{f_{i-1}} M_i \rightarrow^{f_i} M_{i+1} \rightarrow …​ \] of \(R\)-modules and \(R\)-module homomorphisms is exact at \(M_i\) if \(im f_{i-1} = ker f_i\). The sequence is exact if it is exact at every \(M_i\). For example, \[ 0 \rightarrow M' \rightarrow^f M \] is exact if and only if \(f\) is injective. \[ M' \rightarrow^g M \rightarrow 0 \] is exact if and only if \(g\) is surjective. \[ 0 \rightarrow M' \rightarrow^f M \rightarrow^g M'' \rightarrow 0 \] is exact if and only if \(f\) is injective, \(g\) is surjective and \(im f = ker g\), that is \(coker f \cong M''\). We can think of \(M\) as an extension of \(M'\) by \(M''\). A sequence of this form is called a short exact sequence. Exact sequences that are infinite in both directions are called long exact sequences.

Suppose we have a long exact sequence \[ …​ \rightarrow M_{i-1} \rightarrow^{f_{i-1}} M_i \rightarrow^{f_i} M_{i+1} \rightarrow …​ \] Set \(N_{i+1} = im f_i = ker f_{i+1}\). Then we may form short exact sequences \(0 \rightarrow N_i \rightarrow M_i \rightarrow N_{i+1} \rightarrow 0\). Conversely, given short exact sequences of this type, we can form a long exact sequence.

Theorem:

1. \[ M' \rightarrow^u M \rightarrow^v M'' \rightarrow 0 \] is exact \(\iff\) for all \(R\)-modules \(N\) \[ 0 \rightarrow Hom(M'',N) \rightarrow^\bar{v} Hom(M,N) \rightarrow^\bar{u} Hom(M',N) \] is exact.

2. \[ 0 \rightarrow N' \rightarrow^u \rightarrow^v N'' \] is exact \(\iff\) for all \(R\)-modules \(M\) \[ 0\rightarrow Hom(M,N') \rightarrow^\bar{u} Hom(M,N) \rightarrow^\bar{v} Hom(M,N'') \] is exact.

Proof: exercise.