If a morphism from a commutative absolutely flat ring has integral fibers, does it induce an embedding of spectra?

MathOverflow Asked by Badam Baplan on December 18, 2020

All rings are commutative and unital.

Let $A$ be an absolutely flat ring and $A rightarrow B$ a ring monomorphism with integral fibers (i.e. for each $mathfrak{p} in operatorname{Spec}(A), B otimes_A kappa(mathfrak{p})$ is a domain).

Then there is a continuous bijection $operatorname{minSpec}(B) rightarrow operatorname{Spec}(A)$ (where $operatorname{minSpec}$ denotes the space of minimal primes with the subspace topology).

Question: Is it necessarily the case that $operatorname{Spec}(A)$ is homeomorphic to $operatorname{minSpec}(B)$ (the space of minimal primes of $B$)?

Because $operatorname{Spec}(A)$ is compact Hausdorff, this is equivalent to asking if $operatorname{minSpec}(B)$ is necessarily compact.

I suspect the answer is no, but am having difficulty producing a counterexample.

In the following two cases the answer is clearly yes:
(1) If $A$ is a finite product of fields
(2) If $A rightarrow B$ exhibits $B$ as a projective $A$-module.

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