---- > [!definition] Definition. ([[chain complex of modules]]) > Let $R$ be a [[ring]]. A **chain complex** of $R$-[[module|modules]] is a [[sequence]] of $R$-[[module|modules]] and $R$-[[linear map|module homomorphisms]] $\cdots \xrightarrow{d_{i+2}} M_{i+1} \xrightarrow{d_{i+1}} M_{i} \xrightarrow{ \ \ d_{i} \ \ } M_{i-1} \xrightarrow{d_{i-1}} \cdots$ > such that $d_{i} \circ d_{i+1}=0$ (i.e., $\text{im }d_{i+1} \subset \text{ker }d_{i}$) for all $i$. > > The notation $(M_{\bullet}, d_{\bullet})$ is used to denote a complex. A complex may be infinite in both directions — 'tails' of 0's are usually omitted. > >The [[linear map|homomorphisms]] $d_{i}$ are called **boundary maps**, or **differentials**, suggestive of applications in geometry. The modules $M_{i}$ are sometimes called **chain modules** or **chain groups**. > >If $R=\mathbb{Z}$ then $M_{i}$ [[every abelian group is a Z-module, in exactly one way|is just an]] [[abelian group]], and we refer directly to a **chain complex of abelian groups**. > ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```