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.

Ben Lynn blynn@cs.stanford.edu 💡