Homogeneous localization and regularity

Question

Let $k$ be a field, $S = k[x_0,\dots,x_r]$, $I$ a homogeneous ideal of $S$ and $R=S/I$. Let $P$ be a homogeneous prime ideal of $R$ and let $R_{(P)}$ be the homogeneous localization of $R$ at $P$. I seem to have proved that $R_P$ is regular if and only if $R_{(P)}$ is regular. Do you agree? Also, is there any reason to believe that $R_P$ is flat over $R_{(P)}$?

Answer 1

Let $R$ be a noetherian ($\mathbb Z$-)graded ring, and $\mathfrak p\subset R$ a graded prime ideal. Then $R_{\mathfrak p}$ is regular if and only if $R_{(\mathfrak p)}$ is regular.

"$\Leftarrow$" This follows from $\left(R_{(\mathfrak p)}\right)_{\mathfrak pR_{(\mathfrak p)}}=R_{\mathfrak p}$.

"$\Rightarrow$" Now let's suppose one knows the following result (which is part (c) of the exercise 2.2.24 from Bruns and Herzog):

If $(R,\mathfrak m)$ is gr-local then $R$ is regular iff $R_{\mathfrak m}$ is.

The ring $R_{(\mathfrak p)}$ is gr-local with the gr-maximal ideal $\mathfrak pR_{(\mathfrak p)}$, and then it is regular iff $\left(R_{(\mathfrak p)}\right)_{\mathfrak pR_{(\mathfrak p)}}=R_{\mathfrak p}$ is.

Answer 2

This is almost Exercise 2.2.24(b) of Bruns-Herzog. And other parts of Exercise complete this.
Hint.

• Localization is flat.
• $\left(R_{(p)}\right)_{pR_{(p)}}=R_p.$