The Stacks project

Proof. We strongly urge the reader to find their own proof and not read the proof that follows. The statement makes sense by Lemma 88.17.1. To see that it is true assume we have rig-smooth 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

Observe that $H^{-1}(\mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B C)$ and $H^{-1}(\mathop{N\! L}\nolimits _{C/B}^\wedge )$ are annihilated by a power of $I$; this follows from Lemma 88.4.2 part (2) combined with More on Algebra, Lemmas 15.84.6 and 15.84.7 (to deal with the base change by $B \to C$). Hence $H^{-1}(\mathop{N\! L}\nolimits _{C/A}^\wedge )$ is annihilated by a power of $I$. Next, by the characterization of rig-smooth algebras in Lemma 88.4.2 part (2) which in turn refers to More on Algebra, Lemma 15.84.10 part (5) we can choose $f_1, \ldots , f_ s \in IB$ and $g_1, \ldots , g_ t \in IC$ such that $V(f_1, \ldots , f_ s) = V(IB)$ and $V(g_1, \ldots , g_ t) = V(IC)$ and such that $H^0(\mathop{N\! L}\nolimits _{B/A}^\wedge )_{f_ i}$ is a finite projective $B_{f_ i}$-module and $H^0(\mathop{N\! L}\nolimits _{C/B}^\wedge )_{g_ j}$ is a finite projective $C_{g_ j}$-module. Since the cohomologies in degree $-1$ vanish upon localization at $f_ ig_ j$ we get a short exact sequence

and we conclude that $H^0(\mathop{N\! L}\nolimits _{C/A}^\wedge )_{f_ ig_ j}$ is a finite projective $C_{f_ ig_ j}$-module as an extension of same. Thus by the criterion in Lemma 88.4.2 part (2) and via that the criterion in More on Algebra, Lemma 15.84.10 part (4) we conclude that $C$ is rig-smooth over $(A, I)$. $\square$