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