Definition:Exact Sequence

Definition

Let:

$(1): \quad \cdots \longrightarrow M_i \stackrel{d_i}{\longrightarrow} M_{i+1} \stackrel{d_{i+1}}{\longrightarrow} M_{i+2} \stackrel{d_{i+2}}{\longrightarrow} \cdots$

be a sequence of modules $M_i$ and module homomorphisms $d_i$.

• The sequence $(1)$ is null if $d_i \circ d_{i+1} = 0$ for all $i$.

• The sequence $(1)$ is exact if $\operatorname{im} d_i = \ker d_{i+1}$ for all $i$, where $\operatorname{im}$ and $\ker$ denote the image and kernel of mappings repectively.

• The sequence $(1)$ is short if it is finite and has the form

$ 0 \longrightarrow M_2 \stackrel{d_2}{\longrightarrow} M_3 \stackrel{d_3}{\longrightarrow} M_4 \longrightarrow 0$