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How to understand a point of a scheme?
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I am not quite understand what a point of a scheme really is. It's not like the definition of a point in a usual topology, such as Euclidean space. In Hartshorne's, a point seems to be a prime ideal of a given ring, but I am not quite sure. I hope someone can show me a clear picture. Thanks!
algebraic-geometry schemes
asked yesterday
Yuyi Zhang
10416
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up vote
1
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favorite
I am not quite understand what a point of a scheme really is. It's not like the definition of a point in a usual topology, such as Euclidean space. In Hartshorne's, a point seems to be a prime ideal of a given ring, but I am not quite sure. I hope someone can show me a clear picture. Thanks!
algebraic-geometry schemes
asked yesterday
Yuyi Zhang
10416
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am not quite understand what a point of a scheme really is. It's not like the definition of a point in a usual topology, such as Euclidean space. In Hartshorne's, a point seems to be a prime ideal of a given ring, but I am not quite sure. I hope someone can show me a clear picture. Thanks!
algebraic-geometry schemes
asked yesterday
Yuyi Zhang
10416
I am not quite understand what a point of a scheme really is. It's not like the definition of a point in a usual topology, such as Euclidean space. In Hartshorne's, a point seems to be a prime ideal of a given ring, but I am not quite sure. I hope someone can show me a clear picture. Thanks!
algebraic-geometry schemes
algebraic-geometry schemes
asked yesterday
Yuyi Zhang
10416
asked yesterday
Yuyi Zhang
10416
asked yesterday
Yuyi Zhang
10416
asked yesterday
Yuyi Zhang
10416
asked yesterday
Yuyi Zhang
10416
10416
add a comment |
add a comment |
1 Answer
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There are two notions of a point in algebraic geometry. The first one, and the one I think you are most interested in is the following definition.
The first definition
First, let's recall the definition of a scheme. A scheme is a locally ringed space which is locally affine (i.e. locally isomorphic to $newcommandSpec{operatorname{Spec}}Spec A$ for a ring $A$). And as a reminder, a locally ringed space is a topological space, $X$, with a sheaf of rings, $mathcal{O}_X$ whose stalks are local rings.
I've bolded the words topological space, since that part is the relevant part for talking about points of schemes (at least in this first sense). This gives us the definition:
A point of a scheme $(X,mathcal{O}_X)$ is a point of the underlying topological space $X$. Thus points of a scheme are in fact the points of a topological space.
Now you may be wondering, well how do prime ideals enter the picture here? And the answer is, if $A$ is a ring, then the underlying topological space of $Spec A$ is the set of all prime ideals of $A$ with the Zariski topology on them. Thus points of $Spec A$ are precisely the prime ideals of $A$. Since general schemes are locally affine, we can think of the points of a general scheme as being a prime ideal in some affine open set containing that point, however the precise prime ideal depends entirely on the choice of neighborhood of the point and the isomorphism with the spectrum of some ring.
The other definition
If $X$ and $Y$ are schemes, we call a morphism from $Yto X$ a $Y$-valued point of $X$, or if $Y=Spec A$ is affine, we also say it is an $A$-valued point of $X$. The set of $Y$ valued points of a scheme $X$ is then just $operatorname{Hom}(Y,X)$, and the functor $Ymapsto operatorname{Hom}(Y,X)$ is called the functor of points of the scheme $X$, since to a given scheme $Y$ it assigns the set of $Y$-valued points.
I'm not quite sure of the precise history of the usage of the term point for a morphism of schemes, but in many ways it makes sense. For example, if $Y=Spec k$ where $k$ is a field, then a morphism from $Yto X$ is determined by a choice of (topological) point $xin X$ and a choice of map from $kappa(x)to k$, where $kappa(x)$ is the residue field of $X$ at $x$, $mathcal{O}_{X,x}/mathfrak{m}_x$.
Anyway, I think this second definition is not likely to be what you meant, so I won't say too much more on it.
answered 21 hours ago
jgon
9,62111538
1
Later on in the study of algebraic geometry, there is also the notion of a point on a site, which is a functor from the site to Sets which satisfies certain properties. (For example, if the site is a topological space site, then for each point in the underlying set, the functor of taking a sheaf to its stalks at that point is such a functor.)
– Daniel Schepler
21 hours ago
@DanielSchepler Fair point. I'm not too familiar with the details of ag on sites yet, so I didn't think to say anything about those either, but worth noting.
– jgon
21 hours ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
There are two notions of a point in algebraic geometry. The first one, and the one I think you are most interested in is the following definition.
The first definition
First, let's recall the definition of a scheme. A scheme is a locally ringed space which is locally affine (i.e. locally isomorphic to $newcommandSpec{operatorname{Spec}}Spec A$ for a ring $A$). And as a reminder, a locally ringed space is a topological space, $X$, with a sheaf of rings, $mathcal{O}_X$ whose stalks are local rings.
I've bolded the words topological space, since that part is the relevant part for talking about points of schemes (at least in this first sense). This gives us the definition:
A point of a scheme $(X,mathcal{O}_X)$ is a point of the underlying topological space $X$. Thus points of a scheme are in fact the points of a topological space.
Now you may be wondering, well how do prime ideals enter the picture here? And the answer is, if $A$ is a ring, then the underlying topological space of $Spec A$ is the set of all prime ideals of $A$ with the Zariski topology on them. Thus points of $Spec A$ are precisely the prime ideals of $A$. Since general schemes are locally affine, we can think of the points of a general scheme as being a prime ideal in some affine open set containing that point, however the precise prime ideal depends entirely on the choice of neighborhood of the point and the isomorphism with the spectrum of some ring.
The other definition
If $X$ and $Y$ are schemes, we call a morphism from $Yto X$ a $Y$-valued point of $X$, or if $Y=Spec A$ is affine, we also say it is an $A$-valued point of $X$. The set of $Y$ valued points of a scheme $X$ is then just $operatorname{Hom}(Y,X)$, and the functor $Ymapsto operatorname{Hom}(Y,X)$ is called the functor of points of the scheme $X$, since to a given scheme $Y$ it assigns the set of $Y$-valued points.
I'm not quite sure of the precise history of the usage of the term point for a morphism of schemes, but in many ways it makes sense. For example, if $Y=Spec k$ where $k$ is a field, then a morphism from $Yto X$ is determined by a choice of (topological) point $xin X$ and a choice of map from $kappa(x)to k$, where $kappa(x)$ is the residue field of $X$ at $x$, $mathcal{O}_{X,x}/mathfrak{m}_x$.
Anyway, I think this second definition is not likely to be what you meant, so I won't say too much more on it.
answered 21 hours ago
jgon
9,62111538
1
Later on in the study of algebraic geometry, there is also the notion of a point on a site, which is a functor from the site to Sets which satisfies certain properties. (For example, if the site is a topological space site, then for each point in the underlying set, the functor of taking a sheaf to its stalks at that point is such a functor.)
– Daniel Schepler
21 hours ago
@DanielSchepler Fair point. I'm not too familiar with the details of ag on sites yet, so I didn't think to say anything about those either, but worth noting.
– jgon
21 hours ago
add a comment |
up vote
1
down vote
accepted
There are two notions of a point in algebraic geometry. The first one, and the one I think you are most interested in is the following definition.
The first definition
First, let's recall the definition of a scheme. A scheme is a locally ringed space which is locally affine (i.e. locally isomorphic to $newcommandSpec{operatorname{Spec}}Spec A$ for a ring $A$). And as a reminder, a locally ringed space is a topological space, $X$, with a sheaf of rings, $mathcal{O}_X$ whose stalks are local rings.
I've bolded the words topological space, since that part is the relevant part for talking about points of schemes (at least in this first sense). This gives us the definition:
A point of a scheme $(X,mathcal{O}_X)$ is a point of the underlying topological space $X$. Thus points of a scheme are in fact the points of a topological space.
Now you may be wondering, well how do prime ideals enter the picture here? And the answer is, if $A$ is a ring, then the underlying topological space of $Spec A$ is the set of all prime ideals of $A$ with the Zariski topology on them. Thus points of $Spec A$ are precisely the prime ideals of $A$. Since general schemes are locally affine, we can think of the points of a general scheme as being a prime ideal in some affine open set containing that point, however the precise prime ideal depends entirely on the choice of neighborhood of the point and the isomorphism with the spectrum of some ring.
The other definition
If $X$ and $Y$ are schemes, we call a morphism from $Yto X$ a $Y$-valued point of $X$, or if $Y=Spec A$ is affine, we also say it is an $A$-valued point of $X$. The set of $Y$ valued points of a scheme $X$ is then just $operatorname{Hom}(Y,X)$, and the functor $Ymapsto operatorname{Hom}(Y,X)$ is called the functor of points of the scheme $X$, since to a given scheme $Y$ it assigns the set of $Y$-valued points.
I'm not quite sure of the precise history of the usage of the term point for a morphism of schemes, but in many ways it makes sense. For example, if $Y=Spec k$ where $k$ is a field, then a morphism from $Yto X$ is determined by a choice of (topological) point $xin X$ and a choice of map from $kappa(x)to k$, where $kappa(x)$ is the residue field of $X$ at $x$, $mathcal{O}_{X,x}/mathfrak{m}_x$.
Anyway, I think this second definition is not likely to be what you meant, so I won't say too much more on it.
answered 21 hours ago
jgon
9,62111538
1
Later on in the study of algebraic geometry, there is also the notion of a point on a site, which is a functor from the site to Sets which satisfies certain properties. (For example, if the site is a topological space site, then for each point in the underlying set, the functor of taking a sheaf to its stalks at that point is such a functor.)
– Daniel Schepler
21 hours ago
@DanielSchepler Fair point. I'm not too familiar with the details of ag on sites yet, so I didn't think to say anything about those either, but worth noting.
– jgon
21 hours ago
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
There are two notions of a point in algebraic geometry. The first one, and the one I think you are most interested in is the following definition.
The first definition
First, let's recall the definition of a scheme. A scheme is a locally ringed space which is locally affine (i.e. locally isomorphic to $newcommandSpec{operatorname{Spec}}Spec A$ for a ring $A$). And as a reminder, a locally ringed space is a topological space, $X$, with a sheaf of rings, $mathcal{O}_X$ whose stalks are local rings.
I've bolded the words topological space, since that part is the relevant part for talking about points of schemes (at least in this first sense). This gives us the definition:
A point of a scheme $(X,mathcal{O}_X)$ is a point of the underlying topological space $X$. Thus points of a scheme are in fact the points of a topological space.
Now you may be wondering, well how do prime ideals enter the picture here? And the answer is, if $A$ is a ring, then the underlying topological space of $Spec A$ is the set of all prime ideals of $A$ with the Zariski topology on them. Thus points of $Spec A$ are precisely the prime ideals of $A$. Since general schemes are locally affine, we can think of the points of a general scheme as being a prime ideal in some affine open set containing that point, however the precise prime ideal depends entirely on the choice of neighborhood of the point and the isomorphism with the spectrum of some ring.
The other definition
If $X$ and $Y$ are schemes, we call a morphism from $Yto X$ a $Y$-valued point of $X$, or if $Y=Spec A$ is affine, we also say it is an $A$-valued point of $X$. The set of $Y$ valued points of a scheme $X$ is then just $operatorname{Hom}(Y,X)$, and the functor $Ymapsto operatorname{Hom}(Y,X)$ is called the functor of points of the scheme $X$, since to a given scheme $Y$ it assigns the set of $Y$-valued points.
I'm not quite sure of the precise history of the usage of the term point for a morphism of schemes, but in many ways it makes sense. For example, if $Y=Spec k$ where $k$ is a field, then a morphism from $Yto X$ is determined by a choice of (topological) point $xin X$ and a choice of map from $kappa(x)to k$, where $kappa(x)$ is the residue field of $X$ at $x$, $mathcal{O}_{X,x}/mathfrak{m}_x$.
Anyway, I think this second definition is not likely to be what you meant, so I won't say too much more on it.
answered 21 hours ago
jgon
9,62111538
There are two notions of a point in algebraic geometry. The first one, and the one I think you are most interested in is the following definition.
The first definition
First, let's recall the definition of a scheme. A scheme is a locally ringed space which is locally affine (i.e. locally isomorphic to $newcommandSpec{operatorname{Spec}}Spec A$ for a ring $A$). And as a reminder, a locally ringed space is a topological space, $X$, with a sheaf of rings, $mathcal{O}_X$ whose stalks are local rings.
I've bolded the words topological space, since that part is the relevant part for talking about points of schemes (at least in this first sense). This gives us the definition:
A point of a scheme $(X,mathcal{O}_X)$ is a point of the underlying topological space $X$. Thus points of a scheme are in fact the points of a topological space.
Now you may be wondering, well how do prime ideals enter the picture here? And the answer is, if $A$ is a ring, then the underlying topological space of $Spec A$ is the set of all prime ideals of $A$ with the Zariski topology on them. Thus points of $Spec A$ are precisely the prime ideals of $A$. Since general schemes are locally affine, we can think of the points of a general scheme as being a prime ideal in some affine open set containing that point, however the precise prime ideal depends entirely on the choice of neighborhood of the point and the isomorphism with the spectrum of some ring.
The other definition
If $X$ and $Y$ are schemes, we call a morphism from $Yto X$ a $Y$-valued point of $X$, or if $Y=Spec A$ is affine, we also say it is an $A$-valued point of $X$. The set of $Y$ valued points of a scheme $X$ is then just $operatorname{Hom}(Y,X)$, and the functor $Ymapsto operatorname{Hom}(Y,X)$ is called the functor of points of the scheme $X$, since to a given scheme $Y$ it assigns the set of $Y$-valued points.
I'm not quite sure of the precise history of the usage of the term point for a morphism of schemes, but in many ways it makes sense. For example, if $Y=Spec k$ where $k$ is a field, then a morphism from $Yto X$ is determined by a choice of (topological) point $xin X$ and a choice of map from $kappa(x)to k$, where $kappa(x)$ is the residue field of $X$ at $x$, $mathcal{O}_{X,x}/mathfrak{m}_x$.
Anyway, I think this second definition is not likely to be what you meant, so I won't say too much more on it.
answered 21 hours ago
jgon
9,62111538
answered 21 hours ago
jgon
9,62111538
answered 21 hours ago
jgon
9,62111538
answered 21 hours ago
jgon
9,62111538
9,62111538
1
Later on in the study of algebraic geometry, there is also the notion of a point on a site, which is a functor from the site to Sets which satisfies certain properties. (For example, if the site is a topological space site, then for each point in the underlying set, the functor of taking a sheaf to its stalks at that point is such a functor.)
– Daniel Schepler
21 hours ago
@DanielSchepler Fair point. I'm not too familiar with the details of ag on sites yet, so I didn't think to say anything about those either, but worth noting.
– jgon
21 hours ago
add a comment |
1
Later on in the study of algebraic geometry, there is also the notion of a point on a site, which is a functor from the site to Sets which satisfies certain properties. (For example, if the site is a topological space site, then for each point in the underlying set, the functor of taking a sheaf to its stalks at that point is such a functor.)
– Daniel Schepler
21 hours ago
@DanielSchepler Fair point. I'm not too familiar with the details of ag on sites yet, so I didn't think to say anything about those either, but worth noting.
– jgon
21 hours ago
1
1
Later on in the study of algebraic geometry, there is also the notion of a point on a site, which is a functor from the site to Sets which satisfies certain properties. (For example, if the site is a topological space site, then for each point in the underlying set, the functor of taking a sheaf to its stalks at that point is such a functor.)
– Daniel Schepler
21 hours ago
Later on in the study of algebraic geometry, there is also the notion of a point on a site, which is a functor from the site to Sets which satisfies certain properties. (For example, if the site is a topological space site, then for each point in the underlying set, the functor of taking a sheaf to its stalks at that point is such a functor.)
– Daniel Schepler
21 hours ago
@DanielSchepler Fair point. I'm not too familiar with the details of ag on sites yet, so I didn't think to say anything about those either, but worth noting.
– jgon
21 hours ago
@DanielSchepler Fair point. I'm not too familiar with the details of ag on sites yet, so I didn't think to say anything about those either, but worth noting.
– jgon
21 hours ago
add a comment |
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