# 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.

*blynn@cs.stanford.edu*💡