The Stacks project

Remark 10.127.12. Suppose that $R \to S$ is a local homomorphism of local rings, which is essentially of finite presentation. Take any system $(\Lambda , \leq )$, $R_\lambda \to S_\lambda $ with the properties listed in Lemma 10.127.10. What may happen is that this is the “wrong” system, namely, it may happen that property (4) of Lemma 10.127.11 is not satisfied. Here is an example. Let $k$ be a field. Consider the ring

\[ R = k[[z, y_1, y_2, \ldots ]]/(y_ i^2 - zy_{i + 1}). \]

Set $S = R/zR$. As system take $\Lambda = \mathbf{N}$ and $R_ n = k[[z, y_1, \ldots , y_ n]]/(\{ y_ i^2 - zy_{i + 1}\} _{i \leq n-1})$ and $S_ n = R_ n/(z, y_ n^2)$. All the maps $S_ n \otimes _{R_ n} R_{n + 1} \to S_{n + 1}$ are not localizations (i.e., isomorphisms in this case) since $1 \otimes y_{n + 1}^2$ maps to zero. If we take instead $S_ n' = R_ n/zR_ n$ then the maps $S'_ n \otimes _{R_ n} R_{n + 1} \to S'_{n + 1}$ are isomorphisms. The moral of this remark is that we do have to be a little careful in choosing the systems.

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