Definition 35.4.5. A ring map $f: R \to S$ is universally injective if it is universally injective as a morphism in $\text{Mod}_ R$.

Comment #7213 by Chemy the Przemysław on

We have at least two notions of "universally injective" for a map of rings $R \to S$ on the Stacks Project:

The first is the one here, as a map of $R$-modules, 08WE. The second is in the sense of schemes: universally topologically injective, or radicial, 01S2.

They are not really the same. It would be good to add a comment, a warning, or a lemma connecting them, somewhere, because now we have a situation that by a double abuse we can identify these two notions(I have just done it...), and this operation is not justified by anything I could find on the Stacks Project.

An example, take a field $k$, then the ring-diagonal $k \to k \times k$ corresponds to a map from two points to an one point. It is surjective, it is flat, thus it is faithfully flat, but not injective, thus not radicial, thus not schematically universally injective, but universally injective as $k$-modules. Or a nontrivial separable extension of fields $L/k$. Nevertheless, under some assumptions they will be the same, I think.

Comment #7214 by on

Given a ring map $A \to B$ consider the two radically different notions "$A \to B$ is an epimorphism of rings" and "$\text{Spec}(B) \to \text{Spec}(A)$ is an epimorphism of schemes". I think people wouldn't often confuse the two notions. Same here I think.

There are also:

• 4 comment(s) on Section 35.4: Descent for universally injective morphisms

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