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Let $R$ be a $\mathbb{Z}$-graded ring, $P$ a homogeneous prime ideal of $R$, and $U$ the multiplicative subset $R \setminus P$. Let $R_{(P)}$ be the degree $0$ component of $R_P$, and let $f$ be an element of degree $1$ in $R$ that is not contained in $P$. Let $Q$ be the image of $P$ in the quotient $R / (f - 1)$. I have shown the following:

  1. $Q$ is a prime ideal of $R / (f - 1)$.
  2. $R_{(f)}$ (the degree $0$ component of $R_f$) is isomorphic to $R / (f - 1)$.
  3. $(R_{(P)})[x, x^{-1}] \cong R_P$.

I need to show that $R_{(P)} \cong (R / (f - 1))_Q$. I think this should somehow follow by combining the previous results, but I just can't see it. Can anyone help here?

stillconfused
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  • Are you able to finish using this prior post and/or this one on the same exercise? – Bill Dubuque Sep 12 '22 at 19:56
  • I've taken a look at both of those - the first only deals with showing that $R / (f - 1) \cong R_{(f)}$ and the second with showing that $Q$ is prime. Maybe I just don't see how the same techniques apply in this case, but I haven't been able to figure out this last piece from those parts... – stillconfused Sep 12 '22 at 20:15

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