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I have been reading Bhatt's notes on perfectoid spaces and I have stumbled upon a fact whose proof I am unable to understand. Specifically, in Remark 4.2.8 Bhatt describes the functor $A\mapsto A_{!!}$ from $R^a$-algebras (almost $R$-algebras) to $R$-algebras which is left adjoint to the almostification functor (which is just the localization functor to the category of almost algebras). Bhatt gives as an exercise the statement that even though $(-)_{!!}$ doesn't preserve flatness, it does preserve faithful flatness.

I have been unable to solve this myself, so I followed the reference to Gabber-Ramero's Almost ring theory, Remark 3.1.3. Their argument is short so I replicate it here:

Let $\phi:A\to B$ be a morphism of almost algebras. Then $\phi$ is a monomorphism iff $\phi_{!!}$ is injective; moreover, $B_{!!}/\operatorname{Im}(A_{!!})\cong B_!/A_!$ is flat over $A_{!!}$ if and only if $B/A$ is flat over $A$, by proposition 2.4.35.

I am able to follow the claims in this argument, but I don't see how it implies faithful flatness. I suspect there is some faithful flatness criterion at play which I am not aware of, but I couldn't find it in the literature I've checked. Could someone explain how exactly one recovers the result from this argument?

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    $\begingroup$ A map $A \to B$ of rings is faithfully flat if and only if $A \to B$ is injective and $B/A$ is $A$-flat. $\endgroup$
    – Anonymous
    Commented Jun 3, 2020 at 12:36
  • $\begingroup$ @Anonymous I've figured the criterion is of this shape. Do you have some reference for it? $\endgroup$
    – Wojowu
    Commented Jun 3, 2020 at 12:48
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    $\begingroup$ Bourbaki Commutative Algebra I, §3, no. 5, Proposition 9. $\endgroup$
    – abx
    Commented Jun 3, 2020 at 12:58
  • $\begingroup$ @abx Thank you, that's precisely it. The proof also seems to adapt easily to the almost context. I need to finally learn that Bourbaki contains every statement you will ever need :) $\endgroup$
    – Wojowu
    Commented Jun 3, 2020 at 13:40

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To get this off the unanswered list, here is an answer provided by Anonymous and abx in the comments.

The proof uses the following characterization of faithful flatness and its analogue for almost modules:

An $A$-algebra $B$ is faithfully flat if and only if the structure map $A\to B$ is injective and the quotient $B/A$ is flat as an $A$-module.

This criterion can be found in Bourbaki's Commutative Algebra, §3, no. 5, Proposition 9.(b). The proof readily generalizes to the almost context.

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