Excellent ring

Definition

Let $R$ be a Noetherian ring. A Noetherian ring $A$ is regular if $A_{\mathfrak{q}}$ is a regular local ring for every prime ideal $\mathfrak{q}$ of $A$. An algebra $A$ over a field $\kappa$ is geometrically regular if $A \otimes_\kappa L$ is regular for every finite field extension $L$ of $\kappa$. The ring $R$ is excellent if:

1. $R$ is universally catenary.
2. $R$ is a G-ring: let $\mathfrak{p}$ be a prime ideal of $R$ and let $\widehat{R_{\mathfrak{p}}}$ be the completion of the localization $R_{\mathfrak{p}}$. Then for every prime ideal $\mathfrak{q}$ of $R_{\mathfrak{p}}$ with residue field $\kappa(\mathfrak{q})$, the ring $\widehat{R_{\mathfrak{p}}} \otimes_{R_{\mathfrak{p}}} \kappa(\mathfrak{q})$ is geometrically regular over $\kappa(\mathfrak{q})$.
3. $R$ is J-2: let $S$ be a finitely generated $R$-algebra. Then the set of prime ideals $\mathfrak{q}$ of $S$ with $S_{\mathfrak{q}}$ regular is open in the spectrum of $S$ with the Zariski topology.