# 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.