Lemma 37.61.15 (09RK)—The Stacks project

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Lemma 37.61.15. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of schemes. Assume

$X$ is locally Noetherian,
$j \circ i$ is a regular immersion, and
$i$ is perfect.

Then $i$ and $j$ are regular immersions.

Proof. Since $X$ (and hence $Y$ ) is locally Noetherian all 4 types of regular immersions agree, and moreover we may check whether a morphism is a regular immersion on the level of local rings, see Divisors, Lemma 31.20.8. Thus the result follows from Divided Power Algebra, Lemma 23.7.5. $\square$

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