Remark 65.7.6. 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.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BBL. Beware of the difference between the letter 'O' and the digit '0'.