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No, not every epimorphism of rings is a composition of localizations and surjections. An epimorphism of commutative rings is the same thing as a monomorphism of affine schemes. Monomorphisms are not o …

If $A$ is noetherian, then every integral epimorphism $f \colon A \to B$ is surjective. This has been proven by Ferrand: Prop. 3.8 in Monomorphismes de schémas noethérien", Exp. 7 in Séminaire Samuel, …

Here's a proof not using André-Quillen homology. It uses that $\varphi\colon A\to B$ is flat with regular fibers (which is the case if $A\to B$ is smooth). Let $\mathfrak{q}\subseteq B$ be a prime id …

If $A$ is arbitrary and $I$ is an ideal of finite type such that $A/I$ is a flat $A$-module, then $V(I)$ is open and closed. In fact, $A/I$ is a finitely presented $A$-algebra and thus $\operatorname …

(1) is by definition. The standard proof of (2) is via descent of étale morphisms along universal submersions (see SGA1, Exp IX, Thm 4.10, or http://stacks.math.columbia.edu/tag/04DY, or my paper "Sub …