I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.

I'm trying to compile a list of non-obvious theorems about morphisms of schemes which are useful for general intuition but whose proofs are not easy/technical. Here are some examples:

Zariski's Main Theorem:Let $Y$ be quasi-compact, separated and $f:X \to Y$ be separated, quasi-finite, finitely presented. Then there is a factorization $X \to Z \to Y$ where the first map is an open immersion and the second is finite.Mnemonic:(quasi-finite)$\sim$(finite)$\circ$(open immersion)

Nagata's compactification theorem:Let $S$ be qcqs and $f:X \to S$ be separated, finite type. Then $X$ densely embeds into a proper $S$-scheme.Mnemonic:non-horrible schemes have compactifications

Temkin's factorization theorem: Let $Y$ be qcqs and $f: X \to Y$ be separated, quasi-compact. Then there's a factorization $X \to Z \to Y$ with the first being affine and the second proper.Mnemonic:(separated + quasicompact) = (proper)$\circ$.(affine)

Chow's lemma:Let $S$ be noetherian and $f: X \to S$ separated finite type. Then there exists a projective, surjective $S$-morphism $\bar{X} \to X$ which is an isomorphism on a dense subset and where $\bar{X} \to S$ is quasi-projective. Moreover $X$ is proper iff $\bar{X}$ is projective, and if $X$ is reduced $\bar{X}$ can be chosen to be so as well.Mnemonic:reasonable schemes have quasi-projective "replacements" and proper schemes have projective "replacements"

Hopefully it's clear now what I'm looking for. All theorems above have very weak assumptions and very satisfying conclusions. These are what I'm after.

locally quasi-finite&flatlfp cover (EGA IV$_4$, 17.16.2). $\endgroup$2more comments