The Stacks project

Lemma 41.7.2. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

  1. $f$ is a closed immersion,

  2. $f$ is a proper monomorphism,

  3. $f$ is proper, unramified, and universally injective,

  4. $f$ is universally closed, unramified, and a monomorphism,

  5. $f$ is universally closed, unramified, and universally injective,

  6. $f$ is universally closed, locally of finite type, and a monomorphism,

  7. $f$ is universally closed, universally injective, locally of finite type, and formally unramified.

Proof. The equivalence of (4) – (7) follows immediately from Lemma 41.7.1.

Let $f : X \to S$ satisfy (6). Then $f$ is separated, see Schemes, Lemma 26.23.3 and has finite fibres. Hence More on Morphisms, Lemma 37.44.1 shows $f$ is finite. Then Morphisms, Lemma 29.44.15 implies $f$ is a closed immersion, i.e., (1) holds.

Note that (1) $\Rightarrow $ (2) because a closed immersion is proper and a monomorphism (Morphisms, Lemma 29.41.6 and Schemes, Lemma 26.23.8). By Lemma 41.7.1 we see that (2) implies (3). It is clear that (3) implies (5). $\square$

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