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While working through a proof of this paper, at the middle of page 46, the author seems to claim the following is true:

Let $A\rightarrow B$ be an etale map of rings. Suppose that for every prime $P\subset A$ we have $$ \kappa(P)\otimes_{A}B=0 $$ where $\kappa(P)$ is the residue field at the prime ideal $P$. Then $B=0$.

The only thing I seem to be able to extract from here is that $PB=B$ for all primes $P$ of $A$, which does not seem enough for any kind of conclusion, since $B$ is not necessarily a finite $A$-module. Of course if we would have some Noetherianity or some projectivity assumptions , perhaps one can then use the connections between the different definitions of rank of a module. But else I don't know how to use that $A\rightarrow B$ is an etale ring map.

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    $\begingroup$ This doesn't use étale at all. If $\mathfrak q \subseteq B$ is a prime, then it has a preimage $\mathfrak p = f^{-1}(\mathfrak q) \subseteq A$. But then $\kappa(\mathfrak p) \otimes_A B$ maps to $\kappa(\mathfrak q)$, contradicting the hypothesis. We conclude that $\mathfrak q$ does not exist, i.e. $B$ is the zero ring. Geometrically, the assumption is that all fibres of $\operatorname{Spec} B \to \operatorname{Spec} A$ are empty, and the conclusion is that $\operatorname{Spec} B$ is empty. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 13, 2019 at 18:10

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