Example 15.90.10. Let $k$ be a field and put

$R = k[f, T_1, T_2, \ldots ]/(fT_1, fT_2 - T_1, fT_3 - T_2, \ldots ).$

Then $(R, f)$ is not a glueing pair because the map $R[f^\infty ] \to R^\wedge [f^\infty ]$ is not injective as the image of $T_1$ is $f$-divisible in $R^\wedge$. For

$R = k[f, T_1, T_2, \ldots ]/(fT_1, f^2T_2, \ldots ),$

the map $R[f^\infty ] \to R^\wedge [f^\infty ]$ is not surjective as the element $T_1 + fT_2 + f^2 T_3 + \ldots$ is not in the image. In particular, by Remark 15.90.9, these are both examples where $R \to R^\wedge$ is not flat.

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