The Stacks project

108.38 A formally étale non-flat ring map

In this section we give a counterexample to the final sentence in [0, Example 19.10.3(i), EGA] (this was not one of the items caught in their later errata lists). Consider $A \to A/J$ for a local ring $A$ and a nonzero proper ideal $J$ such that $J^2 = J$ (so $J$ isn't finitely generated); the valuation ring of an algebraically closed non-archimedean field with $J$ its maximal ideal is a source of such $(A, J)$. These non-flat quotient maps are formally étale. Namely, suppose given a commutative diagram

\[ \xymatrix{ A/J \ar[r] & R/I \\ A \ar[u] \ar[r]^\varphi & R \ar[u] } \]

where $I$ is an ideal of the ring $R$ with $I^2 = 0$. Then $A \to R$ factors uniquely through $A/J$ because

\[ \varphi (J) = \varphi (J^2) \subset (\varphi (J)A)^2 \subset I^2 = 0. \]

Hence this also provides a counterexample to the formally étale case of the “structure theorem” for locally finite type and formally étale morphisms in [IV, Theorem 18.4.6(i), EGA] (but not a counterexample to part (ii), which is what people actually use in practice). The error in the proof of the latter is that the very last step of the proof is to invoke the incorrect [0, Example 19.3.10(i), EGA], which is how the counterexample just mentioned creeps in.

Proof. See discussion above. $\square$


Comments (3)

Comment #972 by on

Something is wrong in the parser. The \cite command in the first and last line are not parsed.


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Your email address will not be published.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 060H. Beware of the difference between the letter 'O' and the digit '0'.