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

$f$ is a closed immersion,

$f$ is a proper monomorphism,

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

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

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

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

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

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