Affine Varieties over separably closed fields
$begingroup$
Let $E=E^{sep}$ be a separably closed field, i.e. any separable polynomial over $E$ has a solution in $E$, and let $Var_E$ be the category of smooth, affine schemes of finite type over $E$.
On the other hand consider the full subcategory $Var_E^{classic}$ of locally ringed spaces $(X,mathcal{O}_X)$, such that $Xsubsetmathbb{A}^n(E)=E^n$ is a zariski-closed subspace.
Then we have a functor
$$Var_Erightarrow Var_E^{classic}, Xmapsto (X(E),mathcal{O}_{X(E)}),$$
where $X(E):=Mor_E(Spec(E),X)$ and the sheaf $mathcal{O}_{X(E)}:=alpha^{-1}mathcal{O}_X$ is the inverse image for the canonical embedding $alpha:X(E)rightarrow X$.
My question is: Is this functor fully faithful?
For algebraically closed fields $E=E^{alg}$, this is true, since the canonical embedding $alpha(X(E))subset X$ is very dense (i.e. the inclusion $X(E)subset X$ induces a bijection on the topologies) and so via sobrification of $X(E)$ (for any topological space Y only containing closed points, the sobrification $sob(Y)$ is a certain topological structure on the set of irreducible subsets of Y, such that the canonical embedding $Ysubset sob(Y)$ is continuous.), we obtain an inverse for morphisms $(X(E),mathcal{O}_{X(E)})rightarrow (Y(E),mathcal{O}_{Y(E)})$.
But for separably closed fields $X(E)subset X$ is only dense and not necessarily very dense, since there might be closed points in $X$, which are not contained in $X(E)$, so I don't know if there is a way to "gain control" over such closed points.
algebraic-geometry field-theory affine-schemes affine-varieties
asked Jan 6 at 13:59
MariusMarius
327110
$endgroup$
add a comment |
$begingroup$
Let $E=E^{sep}$ be a separably closed field, i.e. any separable polynomial over $E$ has a solution in $E$, and let $Var_E$ be the category of smooth, affine schemes of finite type over $E$.
On the other hand consider the full subcategory $Var_E^{classic}$ of locally ringed spaces $(X,mathcal{O}_X)$, such that $Xsubsetmathbb{A}^n(E)=E^n$ is a zariski-closed subspace.
Then we have a functor
$$Var_Erightarrow Var_E^{classic}, Xmapsto (X(E),mathcal{O}_{X(E)}),$$
where $X(E):=Mor_E(Spec(E),X)$ and the sheaf $mathcal{O}_{X(E)}:=alpha^{-1}mathcal{O}_X$ is the inverse image for the canonical embedding $alpha:X(E)rightarrow X$.
My question is: Is this functor fully faithful?
For algebraically closed fields $E=E^{alg}$, this is true, since the canonical embedding $alpha(X(E))subset X$ is very dense (i.e. the inclusion $X(E)subset X$ induces a bijection on the topologies) and so via sobrification of $X(E)$ (for any topological space Y only containing closed points, the sobrification $sob(Y)$ is a certain topological structure on the set of irreducible subsets of Y, such that the canonical embedding $Ysubset sob(Y)$ is continuous.), we obtain an inverse for morphisms $(X(E),mathcal{O}_{X(E)})rightarrow (Y(E),mathcal{O}_{Y(E)})$.
But for separably closed fields $X(E)subset X$ is only dense and not necessarily very dense, since there might be closed points in $X$, which are not contained in $X(E)$, so I don't know if there is a way to "gain control" over such closed points.
algebraic-geometry field-theory affine-schemes affine-varieties
asked Jan 6 at 13:59
MariusMarius
327110
$endgroup$
1
$begingroup$
I believe that what you're calling separable algebraically closed is usually called just separably closed to distinguish it from being a separable and algebraically closed field (over some implicit base field).
$endgroup$
– jgon
Jan 6 at 15:14
add a comment |
$begingroup$
Let $E=E^{sep}$ be a separably closed field, i.e. any separable polynomial over $E$ has a solution in $E$, and let $Var_E$ be the category of smooth, affine schemes of finite type over $E$.
On the other hand consider the full subcategory $Var_E^{classic}$ of locally ringed spaces $(X,mathcal{O}_X)$, such that $Xsubsetmathbb{A}^n(E)=E^n$ is a zariski-closed subspace.
Then we have a functor
$$Var_Erightarrow Var_E^{classic}, Xmapsto (X(E),mathcal{O}_{X(E)}),$$
where $X(E):=Mor_E(Spec(E),X)$ and the sheaf $mathcal{O}_{X(E)}:=alpha^{-1}mathcal{O}_X$ is the inverse image for the canonical embedding $alpha:X(E)rightarrow X$.
My question is: Is this functor fully faithful?
For algebraically closed fields $E=E^{alg}$, this is true, since the canonical embedding $alpha(X(E))subset X$ is very dense (i.e. the inclusion $X(E)subset X$ induces a bijection on the topologies) and so via sobrification of $X(E)$ (for any topological space Y only containing closed points, the sobrification $sob(Y)$ is a certain topological structure on the set of irreducible subsets of Y, such that the canonical embedding $Ysubset sob(Y)$ is continuous.), we obtain an inverse for morphisms $(X(E),mathcal{O}_{X(E)})rightarrow (Y(E),mathcal{O}_{Y(E)})$.
But for separably closed fields $X(E)subset X$ is only dense and not necessarily very dense, since there might be closed points in $X$, which are not contained in $X(E)$, so I don't know if there is a way to "gain control" over such closed points.
algebraic-geometry field-theory affine-schemes affine-varieties
asked Jan 6 at 13:59
MariusMarius
327110
$endgroup$
Let $E=E^{sep}$ be a separably closed field, i.e. any separable polynomial over $E$ has a solution in $E$, and let $Var_E$ be the category of smooth, affine schemes of finite type over $E$.
On the other hand consider the full subcategory $Var_E^{classic}$ of locally ringed spaces $(X,mathcal{O}_X)$, such that $Xsubsetmathbb{A}^n(E)=E^n$ is a zariski-closed subspace.
Then we have a functor
$$Var_Erightarrow Var_E^{classic}, Xmapsto (X(E),mathcal{O}_{X(E)}),$$
where $X(E):=Mor_E(Spec(E),X)$ and the sheaf $mathcal{O}_{X(E)}:=alpha^{-1}mathcal{O}_X$ is the inverse image for the canonical embedding $alpha:X(E)rightarrow X$.
My question is: Is this functor fully faithful?
For algebraically closed fields $E=E^{alg}$, this is true, since the canonical embedding $alpha(X(E))subset X$ is very dense (i.e. the inclusion $X(E)subset X$ induces a bijection on the topologies) and so via sobrification of $X(E)$ (for any topological space Y only containing closed points, the sobrification $sob(Y)$ is a certain topological structure on the set of irreducible subsets of Y, such that the canonical embedding $Ysubset sob(Y)$ is continuous.), we obtain an inverse for morphisms $(X(E),mathcal{O}_{X(E)})rightarrow (Y(E),mathcal{O}_{Y(E)})$.
But for separably closed fields $X(E)subset X$ is only dense and not necessarily very dense, since there might be closed points in $X$, which are not contained in $X(E)$, so I don't know if there is a way to "gain control" over such closed points.
algebraic-geometry field-theory affine-schemes affine-varieties
algebraic-geometry field-theory affine-schemes affine-varieties
asked Jan 6 at 13:59
MariusMarius
327110
asked Jan 6 at 13:59
MariusMarius
327110
edited Jan 6 at 16:30
Marius
asked Jan 6 at 13:59
MariusMarius
327110
asked Jan 6 at 13:59
MariusMarius
327110
asked Jan 6 at 13:59
MariusMarius
327110
327110
1
$begingroup$
I believe that what you're calling separable algebraically closed is usually called just separably closed to distinguish it from being a separable and algebraically closed field (over some implicit base field).
$endgroup$
– jgon
Jan 6 at 15:14
add a comment |
1
$begingroup$
I believe that what you're calling separable algebraically closed is usually called just separably closed to distinguish it from being a separable and algebraically closed field (over some implicit base field).
$endgroup$
– jgon
Jan 6 at 15:14
1
1
$begingroup$
I believe that what you're calling separable algebraically closed is usually called just separably closed to distinguish it from being a separable and algebraically closed field (over some implicit base field).
$endgroup$
– jgon
Jan 6 at 15:14
$begingroup$
I believe that what you're calling separable algebraically closed is usually called just separably closed to distinguish it from being a separable and algebraically closed field (over some implicit base field).
$endgroup$
– jgon
Jan 6 at 15:14
add a comment |
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$begingroup$
I believe that what you're calling separable algebraically closed is usually called just separably closed to distinguish it from being a separable and algebraically closed field (over some implicit base field).
$endgroup$
– jgon
Jan 6 at 15:14