Question

Someone recently asked what the epimorphisms in the category of schemes are; the other day I had been wondering about the similar question: what are the monomorphisms in the category of schemes? I am often frustrated working with schemes because, unlike a lot of other categories, it is not immediate that you have left cancellation of morphisms when you know the underlying map on sets is injective--and I think it must not be true in general, though I don't have an example in mind. Are there nice situations or additional conditions that guarantee that one may safely cancel morphisms of schemes on the left?

Answer 1

In EGA IV, 17.2.6 the following characterization of monomorphisms is given:

Let $f : X \to Y$ be a morphism locally of finite type. Then the following conditions are equivalent:

a) $f$ is a monomorphism.

b) $f$ is radicial and formally unramified.

c) For every $y \in Y$, the fiber $f^{-1}(y)$ is either empty or isomorphic to $\text{Spec}(k(y))$.

Also note that (due to the adjunction) a morphism between affine schemes is a monomorphism (in the category of schemes) if and only if the associated homomorphism of rings is an epimorphism (in the category of rings) and the latter ones can be characterized in many ways. See, for example, this Samuel seminar and this MO discussion. Monomorphisms of noetherian schemes are treated in detail in Exposé 7 by Daniel Ferrand.

Further examples:

1) Immersions are monomorphisms; this follows from the universal property of a closed resp. open immersion.

2) A morphism $X \to Y$ is a monomorphism if and only if the diagonal $X \to X \times_Y X$ is an isomorphism. In particular, every monomorphism is separated.

3) In EGA IV, 18.12.6 it is shown that proper monomorphisms are exactly the closed immersions.