In this brief post, I want to draw attention to an amazing theorem which deserves to be well-known. Probably many readers are familiar with Nagata’s compactification theorem: if is any qcqs scheme and is a separated morphism of finite type, then can be factored as an open embedding followed by a proper morphism. This is a by-now-classical result, and has many applications.
Less well-known, however, is the following result of Temkin (cf. Theorem 1.1.3 here):
Temkin’s Factorization Theorem. Let be any separated morphism of qcqs schemes. Then can be factored as an affine morphism followed by a proper morphism.
Telling other people about this theorem is an amusing experience. Invariably, their first reaction is that it simply cannot be true, and that the inclusion map should give a counterexample. But then they realize (or I point out) that can be factored as , where is the blowup of at the origin and is the natural (affine!) open immersion of into . Then they are convinced.