Stable epimorphisms of commutative rings

Question

Recall that an epimorphism $f : A \to B$ in a category with fiber products is called stable (or universal) if for every morphism $C \to B$ the base change $A \times_B C \to C$ is an epimorphism.

Question. Is there a classification of stable epimorphisms in the category of commutative rings? If not, what can we say about them? What about examples?

Clearly all surjective homomorphisms are stable. I have looked at the "classical" example of a non-surjective epimorphism, the inclusion $\mathbb{Z} \to \mathbb{Q}$, but couldn't find out if it is stable. If $C \to \mathbb{Q}$ is injective, then $C$ is a localization of $\mathbb{Z}$ and it follows that $\mathbb{Z} \cong \mathbb{Z} \times_{\mathbb{Q}} C \to C$ is an epimorphism. If $C$ is a $\mathbb{Q}$-algebra, then $\mathbb{Z} \times_{\mathbb{Q}} C \to C$ is the localization at $\mathbb{Z} \setminus \{0\}$ and hence, again, an epimorphism. But I don't know what happens for arbitrary homomorphisms $C \to \mathbb{Q}$.

PS: A reference for the general theory of epimorphisms of commutative rings is: Séminaire Samuel. Algèbre commutative, 2, 1967-1968: Les épimorphismes d'anneaux, available on nundam. See also this mathoverflow discussion.

Answer 1

I don't know about a classification, but the inclusion $\mathbb{Z}\to\mathbb{Q}$ is stable. There's probably a more conceptual proof, but:

Let $\alpha:C\to\mathbb{Q}$ be a ring homomorphism. Then $\mathbb{Z}\times_\mathbb{Q}C\to C$ is the inclusion $D\to C$, where $$D=\left\{c\in C:\alpha(c)\in\mathbb{Z}\right\}.$$

Let $c\in C$. Then $nc\in D$ for some non-zero integer $n$. Take the smallest positive such $n$ and let $m=\alpha(nc)\in\mathbb{Z}$. By minimality of $n$, $\operatorname{hcf}(m,n)=1$.

But also $mc-nc^2\in D$. So in $C\otimes_DC$, $$mc\otimes1-n(c\otimes c)=(mc-nc^2)\otimes 1=1\otimes(mc-nc^2)=1\otimes mc-n(c\otimes c),$$ and so $mc\otimes 1=1\otimes mc$. But also $nc\otimes 1=1\otimes nc$. Since $\operatorname{hcf}(m,n)=1$, $c\otimes 1=1\otimes c$.

So the multiplication map $C\otimes_DC\to C$ is an isomorphism, which is equivalent to $D\to C$ being an epimorphism.

[By the way, I'm sure you knew this, but there are certainly simple examples of epimorphisms of commutative rings that are not stable. For example, let $k$ be a field, and consider the inclusions $k[t]\to k[t,t^{-1}]$ and $k[t^{-1}]\to k[t,t^{-1}]$. These are both epimorphisms, but $$k[t]\times_{k[t,t^{-1}]}k[t^{-1}]=k,$$ and the inclusions $k\to k[t]$ and $k\to k[t^{-1}]$ are not epimorphisms.]