The Stacks project

Proposition 15.51.5. Let $R$ be a $P$-ring where $P$ satisfies (A), (B), (C), and (D). If $R \to S$ is essentially of finite type then $S$ is a $P$-ring.

Proof. Since being a $P$-ring is a property of the local rings it is clear that a localization of a $P$-ring is a $P$-ring. Conversely, if every localization at a prime is a $P$-ring, then the ring is a $P$-ring. Thus it suffices to show that $S_\mathfrak q$ is a $P$-ring for every finite type $R$-algebra $S$ and every prime $\mathfrak q$ of $S$. Writing $S$ as a quotient of $R[x_1, \ldots , x_ n]$ we see from Lemma 15.51.3 that it suffices to prove that $R[x_1, \ldots , x_ n]$ is a $P$-ring. By induction on $n$ it suffices to prove that $R[x]$ is a $P$-ring. Let $\mathfrak q \subset R[x]$ be a maximal ideal. By Lemma 15.51.4 it suffices to show that the fibres of

\[ R[x]_\mathfrak q \longrightarrow R[x]_\mathfrak q^\wedge \]

have $P$. If $\mathfrak q$ lies over $\mathfrak p \subset R$, then we may replace $R$ by $R_\mathfrak p$. Hence we may assume that $R$ is a Noetherian local $P$-ring with maximal ideal $\mathfrak m$ and that $\mathfrak q \subset R[x]$ lies over $\mathfrak m$. Note that there is a unique prime $\mathfrak q' \subset R^\wedge [x]$ lying over $\mathfrak q$. Consider the diagram

\[ \xymatrix{ R[x]_\mathfrak q^\wedge \ar[r] & (R^\wedge [x]_{\mathfrak q'})^\wedge \\ R[x]_\mathfrak q \ar[r] \ar[u] & R^\wedge [x]_{\mathfrak q'} \ar[u] } \]

Since $R$ is a $P$-ring the fibres of $R[x] \to R^\wedge [x]$ have $P$ because they are base changes of the fibres of $R \to R^\wedge $ by a finitely generated field extension so (A) applies. Hence the fibres of the lower horizontal arrow have $P$ for example by Lemma 15.51.2. The right vertical arrow is regular because $R^\wedge $ is a G-ring (Propositions 15.50.6 and 15.50.10). It follows that the fibres of the composition $R[x]_\mathfrak q \to (R^\wedge [x]_{\mathfrak q'})^\wedge $ have $P$ by (C). Hence the fibres of the left vertical arrow have $P$ by (D) and the proof is complete. $\square$

Comments (0)

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Your email address will not be published.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 0BIV. Beware of the difference between the letter 'O' and the digit '0'.