Mixing
Lemma about quasi-coherent modules
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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!
algebraic-geometry quasicoherent-sheaves
asked Jan 19 at 15:58
solgaleosolgaleo
6911
$endgroup$
add a comment |
$begingroup$
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!
algebraic-geometry quasicoherent-sheaves
asked Jan 19 at 15:58
solgaleosolgaleo
6911
$endgroup$
add a comment |
$begingroup$
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!
algebraic-geometry quasicoherent-sheaves
asked Jan 19 at 15:58
solgaleosolgaleo
6911
$endgroup$
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!
algebraic-geometry quasicoherent-sheaves
algebraic-geometry quasicoherent-sheaves
asked Jan 19 at 15:58
solgaleosolgaleo
6911
asked Jan 19 at 15:58
solgaleosolgaleo
6911
asked Jan 19 at 15:58
solgaleosolgaleo
6911
asked Jan 19 at 15:58
solgaleosolgaleo
6911
asked Jan 19 at 15:58
solgaleosolgaleo
6911
6911
add a comment |
add a comment |
1 Answer
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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 $r:Itimes Jto R$, $r(i,j)=r_{ij}$. The map $mathcal{O}_X^{(J)}tomathcal{O}_X^{(I)}$ is given 'component wise' on the copies of the $mathcal{O}_X$'s by $x_jmapstosum_ialpha(r_{ij})$, where $x_jinmathcal{O}_X^{(j)}subsetmathcal{O}_X^{(J)}$. 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.
answered Jan 20 at 21:18
Ryan KeletiRyan Keleti
1719
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1 Answer
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1 Answer
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$begingroup$
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 $r:Itimes Jto R$, $r(i,j)=r_{ij}$. The map $mathcal{O}_X^{(J)}tomathcal{O}_X^{(I)}$ is given 'component wise' on the copies of the $mathcal{O}_X$'s by $x_jmapstosum_ialpha(r_{ij})$, where $x_jinmathcal{O}_X^{(j)}subsetmathcal{O}_X^{(J)}$. 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.
answered Jan 20 at 21:18
Ryan KeletiRyan Keleti
1719
$endgroup$
add a comment |
$begingroup$
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 $r:Itimes Jto R$, $r(i,j)=r_{ij}$. The map $mathcal{O}_X^{(J)}tomathcal{O}_X^{(I)}$ is given 'component wise' on the copies of the $mathcal{O}_X$'s by $x_jmapstosum_ialpha(r_{ij})$, where $x_jinmathcal{O}_X^{(j)}subsetmathcal{O}_X^{(J)}$. 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.
answered Jan 20 at 21:18
Ryan KeletiRyan Keleti
1719
$endgroup$
add a comment |
$begingroup$
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 $r:Itimes Jto R$, $r(i,j)=r_{ij}$. The map $mathcal{O}_X^{(J)}tomathcal{O}_X^{(I)}$ is given 'component wise' on the copies of the $mathcal{O}_X$'s by $x_jmapstosum_ialpha(r_{ij})$, where $x_jinmathcal{O}_X^{(j)}subsetmathcal{O}_X^{(J)}$. 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.
answered Jan 20 at 21:18
Ryan KeletiRyan Keleti
1719
$endgroup$
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 $r:Itimes Jto R$, $r(i,j)=r_{ij}$. The map $mathcal{O}_X^{(J)}tomathcal{O}_X^{(I)}$ is given 'component wise' on the copies of the $mathcal{O}_X$'s by $x_jmapstosum_ialpha(r_{ij})$, where $x_jinmathcal{O}_X^{(j)}subsetmathcal{O}_X^{(J)}$. 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.
answered Jan 20 at 21:18
Ryan KeletiRyan Keleti
1719
edited Jan 20 at 21:31
answered Jan 20 at 21:18
Ryan KeletiRyan Keleti
1719
answered Jan 20 at 21:18
Ryan KeletiRyan Keleti
1719
answered Jan 20 at 21:18
Ryan KeletiRyan Keleti
1719
1719
add a comment |
add a comment |
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