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

Let $A\rightarrow B$ be an etale map of rings. Then the underlying map $$ \text{Spec}(B)\rightarrow \text{Spec}(A) $$ is open and the complement of the image is the vanishing locus of a

finitelygenerated ideal in $A$.

The fact that the underlying map is open is well-known. Why does the part about the finite generation hold with no Noetherianity assumptions on $A$?