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?

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?