A characterization of Noetherian regular ring

Question

Proposition. Let $A$ be a Noetherian ring. The folliwing fact are equivalent:

1. $A_P$ is a local regular ring for all $P$ prime ideals of $A$
2. $A_{\mathfrak{M}}$ is a local regular ring for all $\mathfrak{M}$ maximal ideals of $A$

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Clearly, (1) implies (2) but I don't know how to prove that (2) implies (1). I know Serre's Theorem: if $A$ is a local Noetherian ring, then $A$ is a local regular ring if and only if $gldim(A)<\infty$; in this case $gldim(A)=dim(A)$.

A consequence of Serre's Theorem is that a local ring $A$ is a local regular ring if and only if $A_P$ is a local regular ring for all $P$ prime ideals of $A$.

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I also know the following characterization of global dimension:

• FACT A) If $A$ is a local Noetherian ring, then $gldim(A)=pd(K)=\max\{d\in\mathbb{N}\mid Tor_d^A(K,K)\ne 0\}$, where $K$ is the residue field of $A$
• FACT B) If $A$ is a Noetherian ring, then $gldim(A)=\sup\{gldim(A_\mathfrak{M})\mid\mathfrak{M}\in Max(A)\}$

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Proof (attempt) of the proposition. Let $A$ be a Noetherian ring such that (2) holds. By Serre's Theorem, we deduce that $gldim(A_\mathfrak{M})<\infty$ for all maximal ideals $\mathfrak{M}$ of $A$. Then, (now danger deduction, I'm not sure of what now follows in bold) by using the fact B we deduce that $gldim(A)<\infty$. Then, by definition of $gldim(A)$ we have that $pd(A_P/PA_P)<\infty$ for all $P$ prime ideal of $A$. But since $A_P/PA_P$ is the residue field of the local ring $A_P$, we can apply the fact A to $A_P$ obtaining that $pd(A_P)=gldim(A_P)$, then $gldim(A_P)<\infty$. By Serre's theorem applied to the local Noetherian ring $A_P$ we deduce that $A_P$ is a local regular ring.

Answer 1

$\newcommand{\fm}{\mathfrak{m}}\newcommand{\fp}{\mathfrak{p}}\DeclareMathOperator{\gld}{gldim}$Assume that $A_{\fm}$ is regular local for all maximal $\fm$. Let $\fp$ be a prime. Pick a maximal $\fm \supseteq \fp$. Then, $A_{\fp}$ is a localisation of $A_{\fm}$. (Why?)

In general, if $S \subset A$ is a multiplicative set, then $$\gld(S^{-1} A) \leqslant \gld(A).$$

Thus, $\gld(A_{\fp}) \leqslant \gld(A_{\fm}) < \infty$ and hence, $A_{\fp}$ is itself regular local.