Example 15.11.14 (Moret-Bailly). Lemma 15.11.13 is wrong if the colimit isn't filtered. For example, if we take the coproduct of the henselian pairs $(\mathbf{Z}_ p, (p))$ and $(\mathbf{Z}_ p, (p))$, then we obtain $(A, pA)$ with $A = \mathbf{Z}_ p \otimes _\mathbf {Z} \mathbf{Z}_ p$. This isn't a henselian pair: $A/pA = \mathbf{F}_ p$ hence if $(A, pA)$ where henselian, then $A$ would have to be local. However, $\mathop{\mathrm{Spec}}(A)$ is disconnected; for example for odd primes $p$ we have the nontrivial idempotent

where $u \in \mathbf{Z}_ p$ is a square root of $1 + p$. Some details omitted.

## Comments (6)

Comment #4907 by Laurent Moret-Bailly on

Comment #4908 by Pieter Belmans on

Comment #5022 by Johan on

Comment #5178 by Johan on

Comment #5400 by Takagi Benseki（高城 辨積） on

Comment #5633 by Johan on

There are also: