----
$R$ is a [[ring]].
> [!definition] Definition. ([[finitely presented module]])
> An $R$-[[module]] $M$ is **finitely presented** if if for some positive integers $m,n$ there is an [[exact sequence]] $R^{\oplus n} \xrightarrow{\varphi} R^{\oplus m} \to M \to 0.$
> Such a [[chain complex of modules|sequence]] is called a **presentation** of $M$.
^definition
> [!note] Remark.
> The portion $R^{\oplus m} \to M \to 0$ of the [[exact sequence]] comes straight out of the discussion of [[submodule generated by a subset|finitely generated modules]]. Then we prepend $R^{\oplus n}$ to obtain the present discussion.
>
> Although it takes a sufficiently reasonable [[ring]] $R$ to make the notions coincide. See [[finitely presented and finitely generated modules coincide over Noetherian rings]].
^note
> [!equivalence]
> [[exact sequence|Exactness]] implies $(R^{\oplus m} \to M) \circ \varphi=0$ such that the [[cokernel of a module homomorphism|universal property of cokernels]] may be invoked as follows:
>
> ```tikz
> \usepackage{tikz-cd}
> \usepackage{amsmath}
> \begin{document}
> % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACUA9YAHR4jTM4AAjABfEGNLpMufIRQBGclVqMWbLr36CmIgLYSpM7HgJEATCur1mrRCACyk6SAyn5RAMzW1dtsQuJnLmSqSKqrYaDnw4MAAeOMAAxhAA1jAATsJifPSZaAAWWMJ8qWAA5sJafAJCwobCAPSlPHGJwFj6OXl0BcWSqjBQFfBEoABmmRD6SGQgOBBIyn7RIL39WEEgUzPL1ItIFsY707OIVgtLiF4nu+fzhxfUAEYwYFBIXvNR9iCBdzO+yuSAALNRCjA6J8HDgAO4QSHQhCAvaIcEg54LOhYBhsQoQdIgCFQmELBFIqAIagMLBgP5QCA4OKfMQUMRAA
> \begin{tikzcd}
> R^{\oplus n} \arrow[r, "\varphi"] \arrow[rr, "0", bend left] & R^{\oplus m} \arrow[r] \arrow[d, two heads] & M \arrow[r] & 0 \\
> & \text{coker }\varphi \cong R^{\oplus m} / \text{im }\varphi \arrow[ru, two heads, dotted, hook] & &
> \end{tikzcd}
> \end{document}
> ```
> This tells us that $M$ is [[finitely presented module|finitely presented]] if and only if it is (up to [[isomorphism]]) the [[cokernel of a module homomorphism|cokernel]] of a [[linear map|homomorphism]] of [[submodule generated by a subset|finitely generated]] *[[free module|free]]* [[module|modules]].
^equivalence
> [!intuition]
> An [[exact sequence]] $R^{\oplus B} \xrightarrow{\varphi} R^{\oplus A} \xtwoheadrightarrow{\psi} M \to 0$
is the data of a [[surjection]] $R^{\oplus A} \to M$ —— that is, a way to write every $m \in M$ as a linear combination of elements in $A$ —— together with a map $R^{\oplus B} \xrightarrow{\varphi} R^{\oplus A}$ satisfying $\frac{R^{\oplus A}}{\im \varphi}\cong M$. As $\varphi$ is a morphism of free modules, $\im \varphi$ is generated by $\varphi(b)$ for $b \in B$.[^1] If we think of each $\varphi(b)$ as a 'relation', then $M$ corresponds to 'quotienting the module generated by $A$ by the relations generated by $b