I am currently faced with a problem which would be solved instantly if the answer to the question in the title was true.
I know this kind of things are generally not true, but every counterexample I've found of non-surjective ring epimorphisms were either:
- not both local or noetherian (e.g. $\mathbb{Z}\rightarrow \mathbb{Q}$)
- not both complete (any noncomplete noetherian local ring into its completion)
Apparently, a lot of ring epimorphisms are compositions of quotients (surjections), localizations and completions. So it seems to me that it is possible that the result in the title is true.
Has anyone ever heard of such a thing?
PS : I've been trying to prove that they're finite ring maps (because a ring map if surjective iff it's a finite epimorphism) but I haven't succeeded ; I thought maybe using Nakayama would help, but I can't get my head around a solution.
PPS : To be clearer : I am in the category of noetherian local complete rings with a fixed residue field, and with morphisms having to induce the identity on the residues fields. My epimorphism is in this category.
Here are two references:
Stacks Project, Tag 0394
Lemma 2.1 of Ferrand's "Monomorphismes de schémas noethériens" link (I learned about this paper via this post).
See also this question (the Artinian case).