The Stacks project

Lemma 86.15.5. Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$. If $A/I$ is Jacobson for some (equivalently any) ideal of definition $I \subset A$ and $\varphi $ is naively rig-flat, then $\varphi $ is rig-flat.

Proof. Assume $\varphi $ is naively rig-flat. We first state some obvious consequences of the assumptions. Namely, let $f \in A$. Then $A, B, A_{\{ f\} }, B_{\{ f\} }$ are Noetherian adic topological rings. The maps $A \to A_{\{ f\} } \to B_{\{ f\} }$ and $A \to B \to B_{\{ f\} }$ are adic and topologically of finite type. The ring maps $A \to A_{\{ f\} }$ and $B \to B_{\{ f\} }$ are flat as compositions of $A \to A_ f$ and $B \to B_ f$ and the completion maps which are flat by Algebra, Lemma 10.97.2. The quotients of each of the rings $A, B, A_{\{ f\} }, B_{\{ f\} }$ by $I$ is of finite type over $A/I$ and hence Jacobson too (Algebra, Proposition 10.35.19).

Let $\mathfrak q' \subset B_{\{ f\} }$ be rig-closed. It suffices to prove that $(B_{\{ f\} })_{\mathfrak q'}$ is flat over $A_{\{ f\} }$, see Lemma 86.15.1. By Lemma 86.14.5 the primes $\mathfrak q \subset B$ and $\mathfrak p' \subset A_{\{ f\} }$ and $\mathfrak p \subset A$ lying under $\mathfrak q'$ are rig-closed. We are going to apply Algebra, Lemma 10.100.2 to the diagram

\[ \xymatrix{ B_\mathfrak q \ar[r] & (B_{\{ f\} })_{\mathfrak q'} \\ A_\mathfrak p \ar[u] \ar[r] & (A_{\{ f\} })_{\mathfrak p'} \ar[u] } \]

with $M = B_\mathfrak q$. The only assumption that hasn't been checked yet is the fact that $\mathfrak p$ generates the maximal ideal of $(A_{\{ f\} })_{\mathfrak p'}$. This follows from Lemma 86.14.8; here we use that $\mathfrak p$ and $\mathfrak p'$ are rig-closed to see that $f$ maps to a unit of $A/\mathfrak p$ (this is the only step in the proof that fails without the Jacobson assumption). Namely, this tells us that $A/\mathfrak p \to A_{\{ f\} }/\mathfrak p'$ is a finite inclusion of local rings (Lemma 86.14.4) and $f$ maps to a unit in the second one. $\square$

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Your email address will not be published.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