The Stacks project

109.41 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

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