The Stacks project

Lemma 86.15.10. The property $P(\varphi )=$“$\varphi $ is rig-flat” on arrows of $\textit{WAdm}^{Noeth}$ is stable under composition as defined in Formal Spaces, Remark 85.17.14.

Proof. The statement makes sense by Lemma 86.15.9. To see that it is true assume we have rig-flat morphisms $A \to B$ and $B \to C$ in $\textit{WAdm}^{Noeth}$. Then $A \to C$ is adic and topologically of finite type by Lemma 86.11.4. To finish the proof we have to show that for all $f \in A$ the map $A_{\{ f\} } \to C_{\{ f\} }$ is naively rig-flat. Since $A_{\{ f\} } \to B_{\{ f\} }$ and $B_{\{ f\} } \to C_{\{ f\} }$ are naively rig-flat, it suffices to show that compositions of naively rig-flat maps are naively rig-flat. This is a consequence of Algebra, Lemma 10.39.4. $\square$

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