The Stacks project

Lemma 86.15.1. Let $\varphi : A \to B$ be a morphism in $\textit{WAdm}^{adic*}$ (Formal Spaces, Section 85.17). 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.

Proof. Follows from the definitions and Algebra, Lemma 10.39.18. $\square$

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