## Exact Sequences

Let $R$ be a ring. A sequence

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,

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

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

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

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.

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

is exact.

2.

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

is exact.

**Proof:** exercise.