Affine Varieties over separably closed fields












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$begingroup$


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.










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$endgroup$








  • 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
















1












$begingroup$


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.










share|cite|improve this question











$endgroup$








  • 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














1












1








1





$begingroup$


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.










share|cite|improve this question











$endgroup$




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






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edited Jan 6 at 16:30







Marius

















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














  • 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










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