## 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?