The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.8.3. Let $(M_ i, \mu _{ij})$ be a system of $R$-modules over the preordered set $I$. Assume that $I$ is directed. The colimit of the system $(M_ i, \mu _{ij})$ is canonically isomorphic to the module $M$ defined as follows:

  1. as a set let

    \[ M = \left(\coprod \nolimits _{i \in I} M_ i\right)/\sim \]

    where for $m \in M_ i$ and $m' \in M_{i'}$ we have

    \[ m \sim m' \Leftrightarrow \mu _{ij}(m) = \mu _{i'j}(m')\text{ for some }j \geq i, i' \]
  2. as an abelian group for $m \in M_ i$ and $m' \in M_{i'}$ we define the sum of the classes of $m$ and $m'$ in $M$ to be the class of $\mu _{ij}(m) + \mu _{i'j}(m')$ where $j \in I$ is any index with $i \leq j$ and $i' \leq j$, and

  3. as an $R$-module define for $m \in M_ i$ and $x \in R$ the product of $x$ and the class of $m$ in $M$ to be the class of $xm$ in $M$.

The canonical maps $\phi _ i : M_ i \to M$ are induced by the canonical maps $M_ i \to \coprod _{i \in I} M_ i$.

Proof. Omitted. Compare with Categories, Section 4.19. $\square$


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