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Localization - D&F, Section 15.4, Exercise 12

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
I am reading Dummit and Foote, Section 15.4 Localization.

Exercise 12 on page 727 reads as follows:

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Let \(\displaystyle R = \mathbb{R}[x,y,z]/(xy - z^2) \), let P be the prime ideal \(\displaystyle P = (\overline{x}, \overline{y})\) generated by the images of x and y in R and let \(\displaystyle R_P \) be the localization of R at P.

Prove that \(\displaystyle P^2 R_P \cap R = (\overline{x}) \) and is strictly larger \(\displaystyle P^2 \).

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I am somewhat overwhelmed by this problem.

I know that we have \(\displaystyle R_P = D^{-1}R \) where D = R - P and that we have a homomorphism \(\displaystyle \pi : \ R \to R_P = D^{-1}R \) where \(\displaystyle \pi (r) = r/1 \).

However I am having trouble getting a clear understanding of the nature of the elements of the problem let alone making a significant start on the problem.

For example, regarding the nature of \(\displaystyle P = (\overline{x}, \overline{y})\) - D&F describe this as the the prime ideal \(\displaystyle P = (\overline{x}, \overline{y})\) generated by the images of x and y in R - but images under what? x and y are both elements of \(\displaystyle \mathbb{R}[x,y,z] \), so again - what mapping and where do the images lie?

I would appreciate some help with the nature of the elements of this exercise - including the elements R, P and \(\displaystyle P^2R_P \cap R \) and some help with making a significant start on the exercise.

Peter