The Stacks project

Proof. The statement makes sense by Lemma 88.19.1. To see that it is true assume we have rig-étale morphisms $A \to B$ and $B \to C$ in $\textit{WAdm}^{Noeth}$. Then we can choose an ideal of definition $I \subset A$ such that the topology on $C$ and $B$ is the $I$-adic topology. By Lemma 88.3.5 we obtain an exact sequence

There exists a $c \geq 0$ such that for all $a \in I$ multiplication by $a^ c$ is zero on $\mathop{N\! L}\nolimits _{B/A}^\wedge $ in $D(B)$ and $\mathop{N\! L}\nolimits _{C/B}^\wedge $ in $D(C)$. Then of course multiplication by $a^ c$ is zero on $\mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B C$ in $D(C)$ too. Hence $H^0(\mathop{N\! L}\nolimits _{B/A}^\wedge ) \otimes _ A C$, $H^0(\mathop{N\! L}\nolimits _{C/B}^\wedge )$, $H^{-1}(\mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B C)$, and $H^{-1}(\mathop{N\! L}\nolimits _{C/B}^\wedge )$ are annihilated by $a^ c$. From the exact sequence we obtain that multiplication by $a^{2c}$ is zero on $H^0(\mathop{N\! L}\nolimits _{C/A}^\wedge )$ and $H^{-1}(\mathop{N\! L}\nolimits _{C/A}^\wedge )$. It follows from Lemma 88.8.2 that $C$ is rig-étale over $(A, I)$ as desired. $\square$