Lemma 37.58.15. 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.

Lemma 37.58.15. 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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