Can one recover an algebraically closed field $k$ from its category of finitely generated $k$-algebras?
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2
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More generally, given just the dots and arrows of a category (no taking sections!) and the information that for some unknown ring $R$ the category is the category of
- Classical varieties over $R$
- Affine schemes over $R$
- Schemes over $R$
- etc.
in what cases is there a unique ring $R$ (up to isomorphism) which produces my category (up to equivalence)?
algebraic-geometry
asked Nov 17 at 21:34
Cory Griffith
783412
add a comment |
up vote
2
down vote
favorite
More generally, given just the dots and arrows of a category (no taking sections!) and the information that for some unknown ring $R$ the category is the category of
- Classical varieties over $R$
- Affine schemes over $R$
- Schemes over $R$
- etc.
in what cases is there a unique ring $R$ (up to isomorphism) which produces my category (up to equivalence)?
algebraic-geometry
asked Nov 17 at 21:34
Cory Griffith
783412
All of the objects in these categories have maps to $operatorname{spec}(R)$, so you recover $R$ by looking at the terminal object. Is that what you were looking for, or are you looking for something more specific?
– Andrew
Nov 17 at 23:42
2
@Andrew: you recover an object called $R$; there's an additional question of how to recover the underlying set and ring structure of $R$ itself.
– Qiaochu Yuan
Nov 17 at 23:48
@Andrew Sorry, no, that's not what I was looking for. You have only dots and arrows, and not the information of the sheaves, so as far as you know your terminal object is only a terminal object.
– Cory Griffith
Nov 18 at 0:37
I'm cross posting this to math overflow: mathoverflow.net/questions/315717/…
– Cory Griffith
Nov 19 at 19:07
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
More generally, given just the dots and arrows of a category (no taking sections!) and the information that for some unknown ring $R$ the category is the category of
- Classical varieties over $R$
- Affine schemes over $R$
- Schemes over $R$
- etc.
in what cases is there a unique ring $R$ (up to isomorphism) which produces my category (up to equivalence)?
algebraic-geometry
asked Nov 17 at 21:34
Cory Griffith
783412
More generally, given just the dots and arrows of a category (no taking sections!) and the information that for some unknown ring $R$ the category is the category of
- Classical varieties over $R$
- Affine schemes over $R$
- Schemes over $R$
- etc.
in what cases is there a unique ring $R$ (up to isomorphism) which produces my category (up to equivalence)?
algebraic-geometry
algebraic-geometry
asked Nov 17 at 21:34
Cory Griffith
783412
asked Nov 17 at 21:34
Cory Griffith
783412
asked Nov 17 at 21:34
Cory Griffith
783412
asked Nov 17 at 21:34
Cory Griffith
783412
asked Nov 17 at 21:34
Cory Griffith
783412
783412
All of the objects in these categories have maps to $operatorname{spec}(R)$, so you recover $R$ by looking at the terminal object. Is that what you were looking for, or are you looking for something more specific?
– Andrew
Nov 17 at 23:42
2
@Andrew: you recover an object called $R$; there's an additional question of how to recover the underlying set and ring structure of $R$ itself.
– Qiaochu Yuan
Nov 17 at 23:48
@Andrew Sorry, no, that's not what I was looking for. You have only dots and arrows, and not the information of the sheaves, so as far as you know your terminal object is only a terminal object.
– Cory Griffith
Nov 18 at 0:37
I'm cross posting this to math overflow: mathoverflow.net/questions/315717/…
– Cory Griffith
Nov 19 at 19:07
add a comment |
All of the objects in these categories have maps to $operatorname{spec}(R)$, so you recover $R$ by looking at the terminal object. Is that what you were looking for, or are you looking for something more specific?
– Andrew
Nov 17 at 23:42
2
@Andrew: you recover an object called $R$; there's an additional question of how to recover the underlying set and ring structure of $R$ itself.
– Qiaochu Yuan
Nov 17 at 23:48
@Andrew Sorry, no, that's not what I was looking for. You have only dots and arrows, and not the information of the sheaves, so as far as you know your terminal object is only a terminal object.
– Cory Griffith
Nov 18 at 0:37
I'm cross posting this to math overflow: mathoverflow.net/questions/315717/…
– Cory Griffith
Nov 19 at 19:07
All of the objects in these categories have maps to $operatorname{spec}(R)$, so you recover $R$ by looking at the terminal object. Is that what you were looking for, or are you looking for something more specific?
– Andrew
Nov 17 at 23:42
All of the objects in these categories have maps to $operatorname{spec}(R)$, so you recover $R$ by looking at the terminal object. Is that what you were looking for, or are you looking for something more specific?
– Andrew
Nov 17 at 23:42
2
2
@Andrew: you recover an object called $R$; there's an additional question of how to recover the underlying set and ring structure of $R$ itself.
– Qiaochu Yuan
Nov 17 at 23:48
@Andrew: you recover an object called $R$; there's an additional question of how to recover the underlying set and ring structure of $R$ itself.
– Qiaochu Yuan
Nov 17 at 23:48
@Andrew Sorry, no, that's not what I was looking for. You have only dots and arrows, and not the information of the sheaves, so as far as you know your terminal object is only a terminal object.
– Cory Griffith
Nov 18 at 0:37
@Andrew Sorry, no, that's not what I was looking for. You have only dots and arrows, and not the information of the sheaves, so as far as you know your terminal object is only a terminal object.
– Cory Griffith
Nov 18 at 0:37
I'm cross posting this to math overflow: mathoverflow.net/questions/315717/…
– Cory Griffith
Nov 19 at 19:07
I'm cross posting this to math overflow: mathoverflow.net/questions/315717/…
– Cory Griffith
Nov 19 at 19:07
add a comment |
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All of the objects in these categories have maps to $operatorname{spec}(R)$, so you recover $R$ by looking at the terminal object. Is that what you were looking for, or are you looking for something more specific?
– Andrew
Nov 17 at 23:42
2
@Andrew: you recover an object called $R$; there's an additional question of how to recover the underlying set and ring structure of $R$ itself.
– Qiaochu Yuan
Nov 17 at 23:48
@Andrew Sorry, no, that's not what I was looking for. You have only dots and arrows, and not the information of the sheaves, so as far as you know your terminal object is only a terminal object.
– Cory Griffith
Nov 18 at 0:37
I'm cross posting this to math overflow: mathoverflow.net/questions/315717/…
– Cory Griffith
Nov 19 at 19:07