Epic maps in the category of commutative rings with identity.

Question

Here all rings are assumed to lie in the category $\cal C$ of commutative rings with identity, and ${\cal C} (\ R\ ,\ S\ )$ is the set of all ring homomorphisms $F$ from $R$ to $S$ for which $F(1_R)=F(1_S)$. Then $F\in{\mathcal C}(\ R\ ,\ S)$ induces an $R$-module sructure on $S$. Since the functor $\otimes_R$ is right exact, it is not difficult to see that $$S\otimes_RS \cong S\cong S\otimes_SS {\mathrm{\ as \ }} R{\mathrm{-algebras}} \iff \left(S\left/{\mathrm{image}}(F)\right.\right)\otimes_RS=0.\tag{ *} $$

Intuitively, these equivalent conditions seem to show that, for $s_1,\ s_2\ \in S$, the product $s_1 \cdot s_2\in S$ is determined completely by $F(R)$, and just as for localizations, this should show that the equivalent conditions in $(*)$ should also be equivalent to other special statements.

My question if the following: please give, by a reference or a proof or a counterexample to each of the two directions of the following conjecture: $$ S\otimes_RS\cong S \iff F \mathrm{\ is\ a\ flat\ epimorphism\ in\ the \ category \ \mathcal{C}.} $$

Answer 1

It is a general theorem that in any category, a map $f : A \to B$ is an epimorphism if and only if the diagram

$$ \begin{matrix} A &\to& B \\ \downarrow & & \downarrow \\ B &\to& B\end{matrix} $$

is a pushout square, where the top and left arrows are $f$ and the right and bottom arrows are the identity.

In the category of commutative rings, pushouts are tensor products. That is, the following is a pushout square

$$ \begin{matrix} R &\to& S \\ \downarrow & & \downarrow \\ T &\to& S \otimes_R T\end{matrix} $$

(where the top and left arrows are the maps used to induce an $R$-module structure on $S$ and $T$)

Therefore, $R \to S$ is epimorphic if and only if the multiplication map $S \otimes_R S \to S$ is an isomorphism.

I suspect simply having $S \cong S \otimes_R S$ is not enough.

Answer 2

The conjecture is true. The reference is as follows:

Daniel Lazard: "Epimorphisme plats", Seminaire Samuel, Algebre Commutative, tome 2 (1967-1968), exp. no4, pp. 1-12. "Lemme 1.0" will give you what you need.