The Stacks project

Lemma 88.15.1. Let $\varphi : A \to B$ be a morphism in $\textit{WAdm}^{adic*}$ (Formal Spaces, Section 87.21). Assume $\varphi $ is adic. The following are equivalent:

1. $B_ f$ is flat over $A$ for all topologically nilpotent $f \in A$,

2. $B_ g$ is flat over $A$ for all topologically nilpotent $g \in B$,

3. $B_\mathfrak q$ is flat over $A$ for all primes $\mathfrak q \subset B$ which do not contain an ideal of definition,

4. $B_\mathfrak q$ is flat over $A$ for every rig-closed prime $\mathfrak q \subset B$, and

5. add more here.