Commutative Algebra

Let $R$ be a ring. A sequence

$$...\to M_{i-1}{\to }^{f_{i-1}}{M}_i{\to }^{f_i}{M}_{i+1}\to ...$$

of $R$-modules and $R$-module homomorphisms is exact at $M_i$ if $\mathrm{im}{f}_{i-1}=\ker f_i$. The sequence is exact if it is exact at every $M_i$. For example,

$$0\to M\prime {\to }^{f}M$$

is exact if and only if $f$ is injective.

$$M\prime {\to }^{g}M\to 0$$

is exact if and only if $g$ is surjective.

$$0\to M\prime {\to }^{f}M{\to }^{g}M″\to 0$$

is exact if and only if $f$ is injective, $g$ is surjective and $\mathrm{im}f=\ker g$, that is $\mathrm{coker}f\cong M″$. We can think of $M$ as an extension of $M\prime$ 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

$$...\to M_{i-1}{\to }^{f_{i-1}}{M}_i{\to }^{f_i}{M}_{i+1}\to ...$$

Set $N_{i+1}=\mathrm{im}{f}_i=\ker f_{i+1}$. Then we may form short exact sequences $0\to N_i\to M_i\to N_{i+1}\to 0$. Conversely, given short exact sequences of this type, we can form a long exact sequence.

Theorem:

1.

$$M\prime {\to }^{u}M{\to }^{v}M″\to 0$$

is exact $\iff$ for all $R$-modules $N$

$$0\to \mathrm{Hom}(M″,N){\to }^{\bar{v}}\mathrm{Hom}(M,N){\to }^{\bar{u}}\mathrm{Hom}(M\prime ,N)$$

is exact.

2.

$$0\to N\prime {\to }^u{\to }^{v}N″$$

is exact $\iff$ for all $R$-modules $M$

$$0\to \mathrm{Hom}(M,N\prime ){\to }^{\bar{u}}\mathrm{Hom}(M,N){\to }^{\bar{v}}\mathrm{Hom}(M,N″)$$

is exact.

Proof: exercise.