Skip to main content
MathOverflow

Return to Question

Became Hot Network Question
formatting
Source Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286

Question on Extension Theoremthe extension theorem from Humphrey's BookHumphreys' book on Linear Algebraic Groupslinear algebraic groups

On page 2 of Humphrey'sHumphreys' book "Linear Algebraic Groups""Linear algebraic groups" he presents the "Extension Theorem""extension theorem". I will copy it below:

Extension Theorem Let $R/S$ be an integral extension, $K$ an algebraically closed field. Then any homomorphism $\varphi:S\longrightarrow K$ extends to a homomorphism $\varphi^\prime:R\longrightarrow K$. If $x\in R$, $a\in K$, $\varphi$ can first be extended to a homomorphism $R[x]\longrightarrow K$ sending $x$ to $a$ (then be further extended to $R$, $R$ being integral over $R[x]$), provided $f(x)=0$ implies $f_\varphi(a)=0$ for $f(T)\in S[T]$ ($f_\varphi(T)$ the polynomial over $K$ gotten by applying $\varphi$ to each coefficient of $f(T)$).

Extension theorem. Let $R/S$ be an integral extension, $K$ an algebraically closed field. Then any homomorphism $\varphi:S\longrightarrow K$ extends to a homomorphism $\varphi^\prime:R\longrightarrow K$. If $x\in R$, $a\in K$, $\varphi$ can first be extended to a homomorphism $R[x]\longrightarrow K$ sending $x$ to $a$ (then be further extended to $R$, $R$ being integral over $R[x]$), provided $f(x)=0$ implies $f_\varphi(a)=0$ for $f(T)\in S[T]$ ($f_\varphi(T)$ the polynomial over $K$ gotten by applying $\varphi$ to each coefficient of $f(T)$).

I have two questions.

The first is if there is a misprint here and the two occurrences of $R[x]$ should be in fact $S[x]$ (or I am missing something here)? Isn't $R[x]=R$?

The next question is could the inductive procedure described here by extending from $S$ to $S[x]$ etc.., eventually reaching $R$ be infinite? That is, maybe we may need infinitely many $x_i$ such that $R=S[x_1,x_2,...]$? If so, is the number of generators $x_i$ countably infinite or possibly uncountable?

Question on Extension Theorem from Humphrey's Book on Linear Algebraic Groups

On page 2 of Humphrey's book "Linear Algebraic Groups" he presents the "Extension Theorem". I will copy it below:

Extension Theorem Let $R/S$ be an integral extension, $K$ an algebraically closed field. Then any homomorphism $\varphi:S\longrightarrow K$ extends to a homomorphism $\varphi^\prime:R\longrightarrow K$. If $x\in R$, $a\in K$, $\varphi$ can first be extended to a homomorphism $R[x]\longrightarrow K$ sending $x$ to $a$ (then be further extended to $R$, $R$ being integral over $R[x]$), provided $f(x)=0$ implies $f_\varphi(a)=0$ for $f(T)\in S[T]$ ($f_\varphi(T)$ the polynomial over $K$ gotten by applying $\varphi$ to each coefficient of $f(T)$).

I have two questions.

The first is if there is a misprint here and the two occurrences of $R[x]$ should be in fact $S[x]$ (or I am missing something here)? Isn't $R[x]=R$?

The next question is could the inductive procedure described here by extending from $S$ to $S[x]$ etc.., eventually reaching $R$ be infinite? That is, maybe we may need infinitely many $x_i$ such that $R=S[x_1,x_2,...]$? If so, is the number of generators $x_i$ countably infinite or possibly uncountable?

Question on the extension theorem from Humphreys' book on linear algebraic groups

On page 2 of Humphreys' book "Linear algebraic groups" he presents the "extension theorem". I will copy it below:

Extension theorem. Let $R/S$ be an integral extension, $K$ an algebraically closed field. Then any homomorphism $\varphi:S\longrightarrow K$ extends to a homomorphism $\varphi^\prime:R\longrightarrow K$. If $x\in R$, $a\in K$, $\varphi$ can first be extended to a homomorphism $R[x]\longrightarrow K$ sending $x$ to $a$ (then be further extended to $R$, $R$ being integral over $R[x]$), provided $f(x)=0$ implies $f_\varphi(a)=0$ for $f(T)\in S[T]$ ($f_\varphi(T)$ the polynomial over $K$ gotten by applying $\varphi$ to each coefficient of $f(T)$).

I have two questions.

The first is if there is a misprint here and the two occurrences of $R[x]$ should be in fact $S[x]$ (or I am missing something here)? Isn't $R[x]=R$?

The next question is could the inductive procedure described here by extending from $S$ to $S[x]$ etc.., eventually reaching $R$ be infinite? That is, maybe we may need infinitely many $x_i$ such that $R=S[x_1,x_2,...]$? If so, is the number of generators $x_i$ countably infinite or possibly uncountable?

Source Link
Mr Dumas
Mr Dumas
  • 33
  • 3

Question on Extension Theorem from Humphrey's Book on Linear Algebraic Groups

On page 2 of Humphrey's book "Linear Algebraic Groups" he presents the "Extension Theorem". I will copy it below:

Extension Theorem Let $R/S$ be an integral extension, $K$ an algebraically closed field. Then any homomorphism $\varphi:S\longrightarrow K$ extends to a homomorphism $\varphi^\prime:R\longrightarrow K$. If $x\in R$, $a\in K$, $\varphi$ can first be extended to a homomorphism $R[x]\longrightarrow K$ sending $x$ to $a$ (then be further extended to $R$, $R$ being integral over $R[x]$), provided $f(x)=0$ implies $f_\varphi(a)=0$ for $f(T)\in S[T]$ ($f_\varphi(T)$ the polynomial over $K$ gotten by applying $\varphi$ to each coefficient of $f(T)$).

I have two questions.

The first is if there is a misprint here and the two occurrences of $R[x]$ should be in fact $S[x]$ (or I am missing something here)? Isn't $R[x]=R$?

The next question is could the inductive procedure described here by extending from $S$ to $S[x]$ etc.., eventually reaching $R$ be infinite? That is, maybe we may need infinitely many $x_i$ such that $R=S[x_1,x_2,...]$? If so, is the number of generators $x_i$ countably infinite or possibly uncountable?