---- Let $R$ be a [[ring]]. > [!definition] Definition. ([[(co)homology of a complex|homology of a chain complex]]) > The **$i^{th}$ homology** of a [[chain complex of modules|chain complex]] $M_{\bullet}: \cdots \xrightarrow{d_{i+2}} M_{i+1} \xrightarrow{d_{i+1}} M_{i} \xrightarrow{ \ \ d_{i} \ \ } M_{i-1} \xrightarrow{d_{i-1}} \cdots$ > of $R$-[[module|modules]] is the $R$-[[module]] $H_{i}(M_{\bullet}):= \frac{\text{ker }d_{i}}{\text{im }d_{i+1}}.$ > Elements of $\ker d_{i}$ are often referred to as **cycles** and denoted $Z_{i}(M_{\bullet})$; elements of $\text{im }d_{i+1}$ as **boundaries** and denoted $B_{i}(C_{\bullet})$. > > The $i$th homology $H_{i}$ can be viewed as a [[covariant functor]] $H_{i}(-):\mathsf{Chain}(R\text{-}\mathsf{Mod}) \to R\text{-}\mathsf{Mod}$, [[homomorphism on homology induced by a chain map|sending morphisms]] (chain maps) $C_{\bullet} \xrightarrow{f_{\bullet}} D_{\bullet}$ to $f_{*}:H_{i}(C_{\bullet}) \to H_{i}(D_{\bullet})$, $[x] \mapsto [f_{i}(x)]$. ^definition > [!definition] Definition. ([[(co)homology of a complex|cohomology of a cochain complex]]) > The **$i^{th}$ cohomology** of a [[chain complex of modules|cochain complex]] > $M^{\bullet}: \cdots \xrightarrow{d^{i-2}} M^{i-1} \xrightarrow{d^{i-1}} M^{i} \xrightarrow{\ \ d^{i}\ \ } M^{i+1} \xrightarrow{d^{i+1}} \cdots$ > of $R$-[[module|modules]] is the $R$-[[module]] > $H^{i}(M^{\bullet}):= \frac{\text{ker }d^{i}}{\text{im }d^{i-1}}.$ > Elements of $\ker d^{i}$ are often referred to as **cocycles** and denoted $Z^{i}(M^{\bullet})$; elements of $\text{im }d^{i-1}$ as **coboundaries** and denoted $B^{i}(M^{\bullet})$. > > The $i$th cohomology $H^{i}$ can be viewed as a [[covariant functor]] $H^{i}(-):\mathsf{Cochain}(R\text{-}\mathsf{Mod}) \to R\text{-}\mathsf{Mod}$, [[homomorphism on cohomology induced by a cochain map|sending morphisms]] (cochain maps) $C^{\bullet} \xrightarrow{f^{\bullet}} D^{\bullet}$ to $f^{*}:H^{i}(C^{\bullet}) \to H^{i}(D^{\bullet})$, $[y] \mapsto [f^{i}(y)]$. ^definition > [!note] Remark. > Observe $H_{i}(M_{\bullet})=0 \iff \text{im }d_{i+1}=\text{ker }d_{i} \iff M_{\bullet}\text{ is exact at }M_{i}.$ > Thus the homology modules may be thought of as a way to measure the 'failure of a complex from being exact'. ^note > [!specialization] > In the (very special) case where $M_{\bullet}$ is the [[chain complex of modules|complex]] $0 \xrightarrow{}M_{1} \xrightarrow{\varphi} M_{0} \xrightarrow{} 0,$ > we have $H_{1}(M_{\bullet}) \cong \text{ker }\varphi$ and $H_{0}(M_{\bullet}) \cong \text{coker }\varphi$. Homology may therefore be thought of as a *vast* generalization of the notion of [[kernel of a module homomorphism|kernel]] and [[cokernel of a module homomorphism|cokernel]] in the context of $R$-$\mathsf{Mod}$. ^specialization ---- #### ---- #### 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 > ```