The Stacks project

Remark 66.7.6. Let $P$ be a property of local rings. Assume that for any étale ring map $A \to B$ and $\mathfrak q$ is a prime of $B$ lying over the prime $\mathfrak p$ of $A$, then $P(A_\mathfrak p) \Leftrightarrow P(B_\mathfrak q)$. Then we obtain an étale local property of germs $(U, u)$ of schemes by setting $\mathcal{P}(U, u) = P(\mathcal{O}_{U, u})$. In this situation we will use the terminology “the local ring of $X$ at $x$ has $P$” to mean $X$ has property $\mathcal{P}$ at $x$. Here is a list of such properties $P$:

1. Noetherian, see More on Algebra, Lemma 15.44.1,

2. dimension $d$, see More on Algebra, Lemma 15.44.2,

3. regular, see More on Algebra, Lemma 15.44.3,

4. discrete valuation ring, follows from (2), (3), and Algebra, Lemma 10.119.7,

5. reduced, see More on Algebra, Lemma 15.45.4,

6. normal, see More on Algebra, Lemma 15.45.6,

7. Noetherian and depth $k$, see More on Algebra, Lemma 15.45.8,

8. Noetherian and Cohen-Macaulay, see More on Algebra, Lemma 15.45.9,

9. Noetherian and Gorenstein, see Dualizing Complexes, Lemma 47.21.8.

There are more properties for which this holds, for example G-ring and Nagata. If we every need these we will add them here as well as references to detailed proofs of the corresponding algebra facts.