Hard to believe

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 S is any qcqs scheme and f: X \to S is a separated morphism of finite type, then f 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 f:X \to Y be any separated morphism of qcqs schemes. Then f 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 i: \mathbf{A}^2 - \{ 0,0 \} \to \mathbf{A}^2 should give a counterexample. But then they realize (or I point out) that i can be factored as p \circ j, where p: X \to \mathbf{A}^2 is the blowup of \mathbf{A}^2 at the origin and j is the natural (affine!) open immersion of \mathbf{A}^2 - \{ 0,0 \} into X. Then they are convinced.

Unrelated: JW pointed out to me that I am now a professional writer of appendices. Maybe this should worry me?


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