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What is a $k$-scheme isomorphism?

1

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In Example 2.4.2 of Chapter IV 2 of Hartshorne's Algebraic Geometry, it gives $pi:X=Spec(k[t])to Spec(k)$ a scheme over $k$ and $F:Xto X$ and $Spec(k)to Spec(k)$ the morphism with identity on the topological space and $p$-th power on the structure sheaf. It says the $k$-scheme $X_p$ given by $Fcircpi:Xto Spec(k)$ is isomorphic to $pi:Xto Spec(k)$, but it also says the corresponding ring homomorphism is defined by $k[t]to k[t]$ $tmapsto t^p$, which is not an isomorphism. I'm confused about the notion "isomorphism" here. Could anyone explain it?

algebraic-geometry schemes affine-schemes

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asked Mar 23 at 11:40

Li LiLi Li

431211

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1

$begingroup$

In Example 2.4.2 of Chapter IV 2 of Hartshorne's Algebraic Geometry, it gives $pi:X=Spec(k[t])to Spec(k)$ a scheme over $k$ and $F:Xto X$ and $Spec(k)to Spec(k)$ the morphism with identity on the topological space and $p$-th power on the structure sheaf. It says the $k$-scheme $X_p$ given by $Fcircpi:Xto Spec(k)$ is isomorphic to $pi:Xto Spec(k)$, but it also says the corresponding ring homomorphism is defined by $k[t]to k[t]$ $tmapsto t^p$, which is not an isomorphism. I'm confused about the notion "isomorphism" here. Could anyone explain it?

algebraic-geometry schemes affine-schemes

share|cite|improve this question

asked Mar 23 at 11:40

Li LiLi Li

431211

$endgroup$

add a comment | 

$begingroup$

In Example 2.4.2 of Chapter IV 2 of Hartshorne's Algebraic Geometry, it gives $pi:X=Spec(k[t])to Spec(k)$ a scheme over $k$ and $F:Xto X$ and $Spec(k)to Spec(k)$ the morphism with identity on the topological space and $p$-th power on the structure sheaf. It says the $k$-scheme $X_p$ given by $Fcircpi:Xto Spec(k)$ is isomorphic to $pi:Xto Spec(k)$, but it also says the corresponding ring homomorphism is defined by $k[t]to k[t]$ $tmapsto t^p$, which is not an isomorphism. I'm confused about the notion "isomorphism" here. Could anyone explain it?

algebraic-geometry schemes affine-schemes

share|cite|improve this question

asked Mar 23 at 11:40

Li LiLi Li

431211

$endgroup$

share|cite|improve this question

asked Mar 23 at 11:40

Li LiLi Li

431211

share|cite|improve this question

asked Mar 23 at 11:40

Li LiLi Li

431211

asked Mar 23 at 11:40

Li LiLi Li

431211

asked Mar 23 at 11:40

Li LiLi Li

431211

1 Answer
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2

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The isomorphism refers to the isomorphism of schemes $X_p$ and $X$, but this isomorphism is not given by $F'$. I believe the isomorphism is given by $at^n mapsto a^p t^n$ for $a in k$ extended by linearity. Since $k$ is perfect, the inclusion $k^p subset k$ is indeed equality.

share|cite|improve this answer

answered Mar 23 at 22:05

YoungsuYoungsu

1,888715

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2

$begingroup$

The isomorphism refers to the isomorphism of schemes $X_p$ and $X$, but this isomorphism is not given by $F'$. I believe the isomorphism is given by $at^n mapsto a^p t^n$ for $a in k$ extended by linearity. Since $k$ is perfect, the inclusion $k^p subset k$ is indeed equality.

share|cite|improve this answer

answered Mar 23 at 22:05

YoungsuYoungsu

1,888715

$endgroup$

add a comment | 

2

$begingroup$

The isomorphism refers to the isomorphism of schemes $X_p$ and $X$, but this isomorphism is not given by $F'$. I believe the isomorphism is given by $at^n mapsto a^p t^n$ for $a in k$ extended by linearity. Since $k$ is perfect, the inclusion $k^p subset k$ is indeed equality.

share|cite|improve this answer

answered Mar 23 at 22:05

YoungsuYoungsu

1,888715

$endgroup$

add a comment | 

$begingroup$

The isomorphism refers to the isomorphism of schemes $X_p$ and $X$, but this isomorphism is not given by $F'$. I believe the isomorphism is given by $at^n mapsto a^p t^n$ for $a in k$ extended by linearity. Since $k$ is perfect, the inclusion $k^p subset k$ is indeed equality.

share|cite|improve this answer

answered Mar 23 at 22:05

YoungsuYoungsu

1,888715

$endgroup$

share|cite|improve this answer

answered Mar 23 at 22:05

YoungsuYoungsu

1,888715

answered Mar 23 at 22:05

YoungsuYoungsu

1,888715

answered Mar 23 at 22:05

YoungsuYoungsu

1,888715