Let f : X —> Y be a monomorphism of algebraic spaces. Is f representable (by schemes)? After hitting this with a bunch of standard arguments I was led to the following commutative algebra question:
Question: Let A —> B be a local homomorphism of local rings such that the two maps B —> B ⊗_A B are essentially etale and such that A is their equalizer. Then is the map A —> B essentially etale?
This is a first approximation; I have been unable to find an exact translation of the problem on monomorphisms into algebra. The answer to the question is (I think) yes if A is a local ring of dimension 0, or if A —> B is flat (descent of \’etale ring maps).
By the way, I should mention that the statement on monomorphisms of algebraic spaces is true when the morphism is locally of finite type. Namely, any separated, locally quasi-finite morphism of algebraic spaces is representable (by schemes), see Lemma Tag 0418.