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
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.
Comments (2)
Comment #9812 by Branislav Sobot on
Comment #9813 by Branislav Sobot on