(**Background:** In any category, an *epimorphism* is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, "any two functions on $Y$ that agree on the image of $X$ must agree." Even in categories where you have underlying sets, epimorphisms are not the same as surjections; for example, in the category of Hausdorff topological spaces, $f$ is an epimorphism if its image is dense.)


What do epimorphisms of (say commutative) rings look like? It's easy to verify that for any ideal $I$ in a ring $A$, the quotient map $A\to A/I$ is an epimorphism. It's also not hard to see that if $S\subset A$ is a multiplicative subset, then the localization $A\to S^{-1}A$ is an epimorphism. Here's a proof to whet your appetite.



> 
> If $g,h:S^{-1}A\to B$ are two homomorphisms that agree on $A$, then for any element $s^{-1}a\in S^{-1}A$, we have  
> 
>  $$g(s^{-1}a)=g(s)^{-1}g(a)=h(s)^{-1}h(a)=h(s^{-1}a)$$
> 
> 
> 


Also, if $A\to B\_i$ is a finite collection of epimorphisms, where the $B\_i$ have disjoint support as $A$-modules, then $A\to\prod B\_i$ is an epimorphism.


Is every epimorphism of rings some product of combinations of quotients and localizations? To put it another way, suppose $f: A\to B$ is an epimorphism of rings with no kernel which sends non-units to non-units and such that $B$ has no idempotents. Must $f$ be an isomorphism?



What do epimorphisms of (commutative) rings look like?
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 only embeddings, e.g., any localization is an epimorphism and the corresponding morphism of schemes is not a locally closed embedding.


**Example**: Let $C$ be the nodal affine cubic and let $X$ be its normalization. Pick any point $x$ above the node. Then $X\setminus\{x\}\to C$ is a monomorphism (see Proposition below). The corresponding homomorphism of rings is injective but not a localization.


**Proposition** (EGA IV 17.2.6): Let $f\colon X\to Y$ be a morphism *locally of finite type* between schemes. TFAE:


1. $f$ is a monomorphism.
2. Every fiber of $f$ is either an isomorphism or empty.


**Incorrect remark from 2009**: A *flat* epimorphism $A\to B$ is a localization if $A$ is normal and $\mathbb{Q}$-factorial. This is a result by D. Lazard and P. Samuel. [cf. Lazard, *Autour de la platitude*, IV, Prop 4.5]


**Correction of this remark** (May 2022):


1. A *flat* epimorphism $A\to B$ is a localization if $A$ is a normal noetherian domain with *torsion class group*. The result cited above proves this when $A$ is a Dedekind domain.
2. When $A$ is a Dedekind domain whose class group is not torsion, then there exists a flat epimorphism $A\to B$ of finite presentation (so an open immersion on Spec) which is not a localization [Lazard, *Autour de la platitude*, IV, Prop 4.6]. More generally, one can let $A$ be any normal domain whose Cartier class group is not torsion and let $\operatorname{Spec} B$ be the complement of a Cartier divisor $D$ whose class is not torsion.
3. The results 1-2 are best understood as follows. When $A$ is normal and locally $\mathbb{Q}$-factorial, then monomorphisms $A\to B$ correspond to subsets $U:=\operatorname{Spec} B\subseteq \operatorname{Spec} A$ such that $U$ is the complement of a, possibly infinite, union of irreducible divisors $\{D\_i\}\_{i\in I}$ [Raynaud, *Un critère d’effectivité de descente*, Cor. 2.7]. The complement of a Cartier divisor is always affine so it follows that $U$ is the intersection of the affine open subschemes $U\_J$ where $J\subseteq I$ is finite and $U\_J:=\operatorname{Spec} A\setminus \bigcup\_{i\in J} D\_i$. Equivalently, $B$ is the colimit (union if domain) of rings $B\_J$ such that $A\to B\_J$ is a monomorphism of finite type (an open immersion on Spec). When in addition $A$ has torsion class group, the $B\_J$ are localizations (in one element) and it follows that $B$ is a localization.


**Remark**: There was a [seminar](http://www.numdam.org/issues/SAC_1967-1968__2_) on epimorphisms of rings directed by P. Samuel in 1967-68. Raynaud's paper is part of this as well as articles by Lazard that later went into his thesis *Autour de la platitude*.


