As the title suggests, I wonder if anyone can share some techniques or references for constructing interesting epimorphisms from generic commutative rings. Generic is largely open to interpretation, but my thinking is that the main point is that the constructions shouldn't depend on global properties of the base ring, and relatedly that the discussion doesn't become trivial when specialized to quasi-local rings.
Although this question has its roots in commutative ring theory, scheme theoretic answers would also be welcome if that provides a more natural framework.
Why I'm asking this: For some important classes of rings, the behavior of epimorphisms is highly constrained. A couple of well-known examples: (i) if $R$ is reduced or Noetherian, then epimorphisms from $R$ are surjective if and only if $R$ has Krull dimension $0$; (ii) epimorphisms from Dedekind domains must be injective or surjective and they (more or less) decompose into a flat part and a torsion part;
My take away from these examples is that it might be interesting to study, generally, the definitional power of "epimorphisms from $R$ have property $(X)$" where $(X)$ is some intelligently chosen property of morphisms.
Of course, a vital prerequisite to this is the ability to construct interesting test epimorphisms. I quickly realized that, for all the papers on epimorphisms I've read, I have basically no idea how to produce them in a general setting.
The purpose of this post is to seek out knowledge of how to construct interesting epimorphisms of rings in situations as general as possible.
One situation I keep coming back to is the apparent need to understand how to construct test epimorphisms from quasi-local nonreduced rings of Krull dimension $0$ and quasi-local domains of dimension $1$.
Following is some thought vomit, please don't feel compelled to read this, especially if you are familiar with epimorphisms: The simplest epimorphisms are of course quotients by ideals and localizations at multiplicative subsets. These are in fact "good enough" epimorphisms to mostly understand what happens in the case $(X) = surjective$, as mentioned above.
Another classic source of epimorphisms is to adjoin 'pointwise' inverses to a ring, as $R \rightarrow R[x]/(ax^2-x, xa^2-a)$. The result of adjoining a pointwise inverse to every element of a ring is a Von Neumann Regular ring, sometimes called the universally absolutely flat $R$-algebra for obvious reasons. In general, the epimorphisms resulting from these constructions are not flat without the assumption that the minimal prime spectrum of $R$ is compact.
More sophisticated epimorphisms can be constructed by universally inverting maps between finite projective modules (so-called Cohn or universal localizations). If you can construct interesting projective modules over $R$, then I guess theoretically you can produce interesting (necessarily flat) epimorphisms by this method. Unfortunately this is already a dead-end if $R$ is quasi-local, because then universal localizations will just coincide with localizations at multiplicative sets.
More on flat epimorphisms.... when a ring has weak global dimension $1$ (is locally a valuation domain), then the flat epimorphisms coincide with "overrings," and are easy to understand completely in terms of intersections of "localizations" at primes. (For domains, "overring" of $R$ means a ring intermediate between $R$ and its field of fractions. The appropriate generalization to rings of weak global dimension $1$ would be the "ring of finite fractions," which, loosely speaking, has elements homomorphisms $I \rightarrow R$ for finitely generated faithful ideals of $R$, and then the meaning of "localization" needs to be slightly adapted too).
If we drop the wk. gl. dim. $1$ assumption, then we can still understand flat epimorphisms as Gabriel localizations (and even as Kaplansky ideal transforms in case they are finite type), but I find it becomes difficult to actually construct them with confidence due to the reflexive nature of the defining localizing system, i.e. the filter $\mathfrak{F}$ associated to a flat epimorphism $A \rightarrow B$ is $\mathfrak{F} = \{ I \subseteq A \mid IB = B\}$. Unless a localizing filter is a special instance of a universal localization, it hard hard to predict whether it will produce a flat epimorphism.
Another hope, I thought, might be to employ dominions to construct interesting epimorphisms. Recall that if $A \rightarrow B$ is a morphism of rings, the dominion of $A$ in $B$ is defined as $\operatorname{Dom(A,B)} = \{b \in B \mid b \otimes_A 1 - 1 \otimes_A b = 0 \}$. A morphism $A \rightarrow B$ is epi iff $\operatorname{Dom(A,B)} = B$. In general, the dominion of $A$ in $B$ does not induce an epimorphism $A \rightarrow \operatorname{Dom(A,B)}$ but if we keep taking successive dominions and limit intersections, transfinitely, then eventually the corresponding sequence of rings has to stabilize by cardinality considerations, and we get an epimorphism from $A$ as a result. The issue here is that the resulting epimorphism will usually be trivial unless $B$ is chosen carefully (but how??), and even so I'm not confident that the result of the transfinite construction would be workably transparent.
