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.