# Definition:Null Sequence (Homological Algebra)

(Redirected from Definition:Complex (Homological Algebra))

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## 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.