The Stacks project

Reduced equals R0 plus S1.

Lemma 10.157.3. Let $R$ be a Noetherian ring. The following are equivalent:

  1. $R$ is reduced, and

  2. $R$ has properties $(R_0)$ and $(S_1)$.

Proof. Suppose that $R$ is reduced. Then $R_{\mathfrak p}$ is a field for every minimal prime $\mathfrak p$ of $R$, according to Lemma 10.25.1. Hence we have $(R_0)$. Let $\mathfrak p$ be a prime of height $\geq 1$. Then $A = R_{\mathfrak p}$ is a reduced local ring of dimension $\geq 1$. Hence its maximal ideal $\mathfrak m$ is not an associated prime since this would mean there exists an $x \in \mathfrak m$ with annihilator $\mathfrak m$ so $x^2 = 0$. Hence the depth of $A = R_{\mathfrak p}$ is at least one, by Lemma 10.63.9. This shows that $(S_1)$ holds.

Conversely, assume that $R$ satisfies $(R_0)$ and $(S_1)$. If $\mathfrak p$ is a minimal prime of $R$, then $R_{\mathfrak p}$ is a field by $(R_0)$, and hence is reduced. If $\mathfrak p$ is not minimal, then we see that $R_{\mathfrak p}$ has depth $\geq 1$ by $(S_1)$ and we conclude there exists an element $t \in \mathfrak pR_{\mathfrak p}$ such that $R_{\mathfrak p} \to R_{\mathfrak p}[1/t]$ is injective. Now $R_\mathfrak p[1/t]$ is contained in the product of its localizations at prime ideals, see Lemma 10.23.1. This implies that $R_{\mathfrak p}$ is a subring of a product of localizations of $R$ at $\mathfrak p \supset \mathfrak q$ with $t \not\in \mathfrak q$. Since these primes have smaller height by induction on the height we conclude that $R$ is reduced. $\square$

Comments (6)

Comment #929 by correction_bot on

Hah, bad slogan!

Comment #7081 by Yuto Masamura on

I think that the sentence "This implies that is a subring of localizations of at primes of smaller height" (in the 2nd paragraph of proof) is not trivial. We can find a prime with (here I assume , i.e., without loss of generality) and thus maps , but I do not know how to verify that the map is injective. Or did you mean 'the product of localizations...', i.e., that we have ?

Comment #7258 by on

Yes, I did mean product of localizations. Thanks very much for catching this. I have made some edits in this commit.

Comment #8783 by Mateo on

The inclusion of prime ideals is reversed in the second paragraph.

There are also:

  • 4 comment(s) on Section 10.157: Serre's criterion for normality

Post a comment

Your email address will not be published. Required fields are marked.

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.

All contributions are licensed under the GNU Free Documentation License.

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 031R. Beware of the difference between the letter 'O' and the digit '0'.