In the category of schemes, what conditions on a closed monomorphism make it a closed immersion?

Question

If the question can be simplified, we can work on smaller categories, such as the category of varieties or schemes of finite type over a field, etc.

By a closed morphism of schemes I mean it is closed as a continuous map, not a closed immersion.

Answer 1

Closed immersions are exactly the proper monomorphisms, see Stacks 04KV. You'll also find a few other conditions there. Any of the following conditions in addition to being a monomorphism will imply your morphism is a closed immersion:

• proper (i.e. universally closed, separated, finite type)
• universally closed + unramified
• universally closed + locally of finite type

This implies, for instance, that a universally closed monomorphism of varieties (schemes of finite type over a field) is a closed immersion.