Definition:Null Sequence (Homological Algebra)

Definition

Let $\left({R, +, \cdot}\right)$ be a ring.

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 $R$-modules $M_i$ and $R$-module homomorphisms $d_i$.

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

Also known as

A null sequence is also known as a differential complex, arising from the Correspondence Between Differential Complexes and Null Sequences.

Sometimes just the term complex is used, but this can be ambiguous unless the context is clarified carefully.

Also see

• Definition:Differential Complex
• Correspondence Between Differential Complexes and Null Sequences