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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.

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How about something like Chevalley's theorem? It says that a quasi-compact morphism which is locally of finite type preserves locally constructible sets. – Dylan Wilson Mar 25 at 1:49
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You could add Stein factorization to the list: (proper) = (integral) $\circ$ (proper with connected geometric fibers). stacks.math.columbia.edu/tag/03GX – Marc Hoyois Mar 25 at 2:47
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Open immersion = etale + monic (EGA IV$_4$, 17.9.1+epsilon), integral morphism = affine + universally closed (EGA IV$_4$, 18.12.8), closed immersion = proper monomorphism (EGA IV$_4$, 18.12.6), qcqs = relatively affine over finite type Z-scheme (Thomason-Trobaugh, Theorem C.9), relative ampleness for a proper fp map is fibral (EGA IV$_3$, 9.6.5), lfp map is unramifed iff etale-locally a closed immersion (EGA IV$_4$, 17.7.1, 18.8.3), qc separated map is immersion iff monic + Mochizuki's valuative criterion (Ch. I, sec. 2.4 of his book on p-adic Teichmuller theory in noetherian case). – nfdc23 Mar 25 at 7:46
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Finitely presented flat map is finite iff quasi-finite & separated with locally constant fiber rank (II, 1.19 Deligne-Rapoport in noetherian case), lfp = commutes with "expected" limits (EGA IV$_3$, 8.14.2), valuative criterion for flatness over a reduced base (EGA IV$_3$, 11.8.1 in noetherian case with dvr's, 4.2.10 in Part I of Raynaud-Gruson in general), universal schematic dominance/density is fibral under "expected" flatness hypotheses (EGA IV$_3$, 11.10.9, 11.10.10), any flat lfp cover admits a section over a locally quasi-finite & flat lfp cover (EGA IV$_4$, 17.16.2). – nfdc23 Mar 25 at 8:35
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Somoth/étale morphisms can be lifted (Zariski-locally on the source) along closed immersions (EGA IV_4, 18.1.1). – Adeel Khan Mar 25 at 16:48

Here's a theorem I find useful:

Theorem. Let $\phi \colon X \to Y$ be a smooth morphism of schemes of relative dimension $d$. Then there exists an open cover $X = \bigcup U_i$ of $X$ such that each $U_i \to Y$ factors as $$U_i \stackrel \pi \to \mathbb A^d_Y \to Y,$$ with $\pi$ étale. Mnemonic: smooth morphisms have étale coordinates.

See Tag 054L.

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EGA 4 vol. 4 Corollary 17.11.4 expresses this as a necessary and sufficient criterion for a map to be smooth at a point. – S. Carnahan Mar 25 at 6:53
    
Shouldn't the $d_i$'s be constant here? – Saal Hardali Mar 25 at 12:41
    
@SaalHardali: they are (automatically) locally constant. – R. van Dobben de Bruyn Mar 25 at 21:49
    
@R.vanDobbendeBruyn Yes that's what I meant. Just wanted to make sure. Thanks. – Saal Hardali Mar 25 at 21:50
    
@R.vanDobbendeBruyn Under the same conditions, how likely would you say is the existence of a factorization $X \to E \to Y$ where the first map is etale and $E$ is some vector bundle over $Y$? – Saal Hardali Mar 27 at 1:45

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