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Thread: A Certain Integral Closure

  1. #1
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    A Certain Integral Closure

    I have a bunch of assertions without motivation which I'm trying to sort out.

    Let H be a subgroup of S_n, [; A = k[x_1,...,x_n] ;], and [; \sigma_i ;] the elementary symmetric polynomials. The assertions are:

    i) A_H is defined as the integral closure of [; k[\sigma_1,...,\sigma_n] ;] in [; k(x_1,...,x_n)^H ;], and [; A_H = k[x_1,...,x_n] \cap k(x_1,...,x_n)^H ;].

    ii) [; k(x_1,...,x_n)^H ;] is the field of fractions of A_H, "i.e." [; k(x_1,...,x_n)^H = A_H[1/k[\sigma_1,...,\sigma_n]] ;].

    Alright, so about the first assertion, that A_H is the intersection. This intersection contains only polynomials, i.e. we must have [; k[x_1,...,x_n] \cap k(x_1,...,x_n)^H \subset k[x_1,...,x_n] ;], so we can simply consider the intersection [; k[x_1,...,x_n] \cap k[x_1,...,x_n]^H ;], which must be [; k[x_1,...,x_n]^H ;]. I've no idea how to show or see that [; k[x_1,...,x_n]^H ;] is the supposed integral closure, but let's leave it at that for the moment.

    Second assertion. What the notation [; A_H[1/k[\sigma_1,...,\sigma_n]] ;] means I have no idea; my guess is that it's the set [; \{ f/g \mid f \in A_H, g \in k[\sigma_1,...,\sigma_n] \} ;]. If this is the case, how do I verify it? Because I would have thought that Frac A_H = k(x_1,...,x_n)^H?
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  2. #2
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    Re: A Certain Integral Closure

    Quote Originally Posted by spudwish View Post
    I have a bunch of assertions without motivation which I'm trying to sort out.

    Let H be a subgroup of S_n, $\displaystyle A = k[x_1,...,x_n] $, and $\displaystyle \sigma_i $ the elementary symmetric polynomials. The assertions are:

    i) A_H is defined as the integral closure of $\displaystyle k[\sigma_1,...,\sigma_n] $ in $\displaystyle k(x_1,...,x_n)^H $, and $\displaystyle A_H = k[x_1,...,x_n] \cap k(x_1,...,x_n)^H $.

    ii) $\displaystyle k(x_1,...,x_n)^H $ is the field of fractions of A_H, "i.e." $\displaystyle k(x_1,...,x_n)^H = A_H[1/k[\sigma_1,...,\sigma_n]] $.

    Alright, so about the first assertion, that A_H is the intersection. This intersection contains only polynomials, i.e. we must have $\displaystyle k[x_1,...,x_n] \cap k(x_1,...,x_n)^H \subset k[x_1,...,x_n] $, so we can simply consider the intersection $\displaystyle k[x_1,...,x_n] \cap k[x_1,...,x_n]^H $, which must be $\displaystyle k[x_1,...,x_n]^H $. I've no idea how to show or see that $\displaystyle k[x_1,...,x_n]^H $ is the supposed integral closure, but let's leave it at that for the moment.

    Second assertion. What the notation $\displaystyle A_H[1/k[\sigma_1,...,\sigma_n]] $ means I have no idea; my guess is that it's the set $\displaystyle \{ f/g \mid f \in A_H, g \in k[\sigma_1,...,\sigma_n] \} $. If this is the case, how do I verify it? Because I would have thought that Frac A_H = k(x_1,...,x_n)^H?
    I have replaced your [; and ;] with "tex" and "/tex".
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