# Etale map has image whose complement is the vanishing locus of a finitely generated ideal

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 finitely generated 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 $$A$$?

Let $$U$$ be the image of $$\operatorname{Spec} B$$ in $$\operatorname{Spec} A$$. Since the map $$\operatorname{Spec} B \to \operatorname{Spec} A$$ is open, $$U$$ is open. And since $$\operatorname{Spec} B$$ is quasi-compact, $$U$$ is quasi-compact. Therefore, $$U$$ is the union of finitely many basic opens in $$\operatorname{Spec} A$$, i.e. there exist $$f_1, \ldots, f_n \in A$$ such that $$U = \bigcup_{i = 1}^n\operatorname{Spec} A[{1}/{f_i}]$$ as subsets of $$\operatorname{Spec} A$$. Then the complement of $$U$$ in $$\operatorname{Spec} A$$ is exactly the vanishing locus of the finitely generated ideal $$I = (f_1, \ldots, f_n)$$.