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Defining abstract varieties and their morphisms over a finitely generated subfield of the base field
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Let $k$ be an algebraically closed field.
By a finitely generated subfield of $k$ I mean a subfield $k_0subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $mathbb Q$ or $mathbb F_p$).
I need a reference or a proof for the following (well-known? evident?) proposition:
Proposition.
Let
$$fcolon Xto V$$
be a morphism of $k$-varieties.
Then the triple $(X,Y,f)$ can be defined over a finitely generated subfield of $k$.
In other words, there exists a finitely generated subfield $k_0subset k$
and a morphism of $k_0$-varieties
$$f_0colon X_0to Y_0$$
such that
$(X_0,Y_0,f_0)times_k_0 k$ is isomorphic to $(X,Y,f)$.
A referee asked me to prove this assertion. I know how to prove this for affine and projective varieties, but not for abstract varieties.
ag.algebraic-geometry arithmetic-geometry schemes
asked 1 hour ago
Mikhail Borovoi
5,6541641
add a comment |Â
up vote
3
down vote
favorite
Let $k$ be an algebraically closed field.
By a finitely generated subfield of $k$ I mean a subfield $k_0subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $mathbb Q$ or $mathbb F_p$).
I need a reference or a proof for the following (well-known? evident?) proposition:
Proposition.
Let
$$fcolon Xto V$$
be a morphism of $k$-varieties.
Then the triple $(X,Y,f)$ can be defined over a finitely generated subfield of $k$.
In other words, there exists a finitely generated subfield $k_0subset k$
and a morphism of $k_0$-varieties
$$f_0colon X_0to Y_0$$
such that
$(X_0,Y_0,f_0)times_k_0 k$ is isomorphic to $(X,Y,f)$.
A referee asked me to prove this assertion. I know how to prove this for affine and projective varieties, but not for abstract varieties.
ag.algebraic-geometry arithmetic-geometry schemes
asked 1 hour ago
Mikhail Borovoi
5,6541641
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let $k$ be an algebraically closed field.
By a finitely generated subfield of $k$ I mean a subfield $k_0subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $mathbb Q$ or $mathbb F_p$).
I need a reference or a proof for the following (well-known? evident?) proposition:
Proposition.
Let
$$fcolon Xto V$$
be a morphism of $k$-varieties.
Then the triple $(X,Y,f)$ can be defined over a finitely generated subfield of $k$.
In other words, there exists a finitely generated subfield $k_0subset k$
and a morphism of $k_0$-varieties
$$f_0colon X_0to Y_0$$
such that
$(X_0,Y_0,f_0)times_k_0 k$ is isomorphic to $(X,Y,f)$.
A referee asked me to prove this assertion. I know how to prove this for affine and projective varieties, but not for abstract varieties.
ag.algebraic-geometry arithmetic-geometry schemes
asked 1 hour ago
Mikhail Borovoi
5,6541641
Let $k$ be an algebraically closed field.
By a finitely generated subfield of $k$ I mean a subfield $k_0subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $mathbb Q$ or $mathbb F_p$).
I need a reference or a proof for the following (well-known? evident?) proposition:
Proposition.
Let
$$fcolon Xto V$$
be a morphism of $k$-varieties.
Then the triple $(X,Y,f)$ can be defined over a finitely generated subfield of $k$.
In other words, there exists a finitely generated subfield $k_0subset k$
and a morphism of $k_0$-varieties
$$f_0colon X_0to Y_0$$
such that
$(X_0,Y_0,f_0)times_k_0 k$ is isomorphic to $(X,Y,f)$.
A referee asked me to prove this assertion. I know how to prove this for affine and projective varieties, but not for abstract varieties.
ag.algebraic-geometry arithmetic-geometry schemes
ag.algebraic-geometry arithmetic-geometry schemes
asked 1 hour ago
Mikhail Borovoi
5,6541641
asked 1 hour ago
Mikhail Borovoi
5,6541641
asked 1 hour ago
Mikhail Borovoi
5,6541641
asked 1 hour ago
Mikhail Borovoi
5,6541641
asked 1 hour ago
Mikhail Borovoi
5,6541641
5,6541641
add a comment |Â
add a comment |Â
1 Answer
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up vote
6
down vote
This is treated (in much greater generality) in EGA IV$_3$, Théorème 8.8.2. The existence of $X_0$ and $Y_0$ follows from part (ii), whereas the existence of $f_0$ is part (i).
References.
[EGA IV$_3$] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie). Publ. Math., Inst. Hautes Étud. Sci. 28, 1-255 (1966). ZBL0144.19904. DOI: 10.1007/BF02684343
answered 54 mins ago
R. van Dobben de Bruyn
9,38122858
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
This is treated (in much greater generality) in EGA IV$_3$, Théorème 8.8.2. The existence of $X_0$ and $Y_0$ follows from part (ii), whereas the existence of $f_0$ is part (i).
References.
[EGA IV$_3$] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie). Publ. Math., Inst. Hautes Étud. Sci. 28, 1-255 (1966). ZBL0144.19904. DOI: 10.1007/BF02684343
answered 54 mins ago
R. van Dobben de Bruyn
9,38122858
add a comment |Â
up vote
6
down vote
This is treated (in much greater generality) in EGA IV$_3$, Théorème 8.8.2. The existence of $X_0$ and $Y_0$ follows from part (ii), whereas the existence of $f_0$ is part (i).
References.
[EGA IV$_3$] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie). Publ. Math., Inst. Hautes Étud. Sci. 28, 1-255 (1966). ZBL0144.19904. DOI: 10.1007/BF02684343
answered 54 mins ago
R. van Dobben de Bruyn
9,38122858
add a comment |Â
up vote
6
down vote
up vote
6
down vote
This is treated (in much greater generality) in EGA IV$_3$, Théorème 8.8.2. The existence of $X_0$ and $Y_0$ follows from part (ii), whereas the existence of $f_0$ is part (i).
References.
[EGA IV$_3$] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie). Publ. Math., Inst. Hautes Étud. Sci. 28, 1-255 (1966). ZBL0144.19904. DOI: 10.1007/BF02684343
answered 54 mins ago
R. van Dobben de Bruyn
9,38122858
This is treated (in much greater generality) in EGA IV$_3$, Théorème 8.8.2. The existence of $X_0$ and $Y_0$ follows from part (ii), whereas the existence of $f_0$ is part (i).
References.
[EGA IV$_3$] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie). Publ. Math., Inst. Hautes Étud. Sci. 28, 1-255 (1966). ZBL0144.19904. DOI: 10.1007/BF02684343
answered 54 mins ago
R. van Dobben de Bruyn
9,38122858
answered 54 mins ago
R. van Dobben de Bruyn
9,38122858
answered 54 mins ago
R. van Dobben de Bruyn
9,38122858
answered 54 mins ago
R. van Dobben de Bruyn
9,38122858
9,38122858
add a comment |Â
add a comment |Â
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