I am trying to understand quasi-coherent modules from https://stacks.math.columbia.edu/tag/01BD . Let me state partially the lemma 17.10.5 about the construction of an example of quasi-coherent module.
Let $(X,\mathcal{O}_X)$ be ringed space. Let $α:R→Γ(X,\mathcal{O}_X)$ be a ring homomorphism from a ring $R$ into the ring of global sections on $X$. Let $M$ be an $R$-module. Choose a presentation $⨁_{j∈J}R→⨁_{i∈I}R→M→0$.
Set $\mathcal{F}_2=Coker(⨁_{j∈J}\mathcal{O}_X→⨁_{i∈I}\mathcal{O}_X)$. Here the map on the component $\mathcal{O}_X$ corresponding to $j∈J$ given by the section $∑_{i}α(r_{ij})$ where the $r_{ij}$ are the matrix coefficients of the map in the presentation of $M$.
It is not clear to me how this map is being defined and especially what is meant by the map of representation.
Thanks in advance!
The map $R^{(J)}\to R^{(I)}$ admits a matrix representation $(r_{ij})$ as $R^{(J)}$ and $R^{(I)}$ are free modules. As $I$ and $J$ may not be finite, what we mean by a matrix here is a map $r:I\times J\to R$, $r(i,j)=r_{ij}$. The map $\mathcal{O}_X^{(J)}\to\mathcal{O}_X^{(I)}$ is given 'component wise' on the copies of the $\mathcal{O}_X$'s by $x_j\mapsto\sum_i\alpha(r_{ij})$, where $x_j\in\mathcal{O}_X^{(j)}\subset\mathcal{O}_X^{(J)}$. Edit: the $x_j$ is determined by $R^{(J)}=\Gamma(X,\mathcal{O}_X^{(J)})$. We take the image of the map under the global sections functor.