Lemma 31.21.7. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of schemes.

1. If $i$ and $j$ are regular immersions, so is $j \circ i$.

2. If $i$ and $j$ are Koszul-regular immersions, so is $j \circ i$.

3. If $i$ and $j$ are $H_1$-regular immersions, so is $j \circ i$.

4. If $i$ is an $H_1$-regular immersion and $j$ is a quasi-regular immersion, then $j \circ i$ is a quasi-regular immersion.

Proof. The algebraic version of (1) is Algebra, Lemma 10.68.7. The algebraic version of (2) is More on Algebra, Lemma 15.30.13. The algebraic version of (3) is More on Algebra, Lemma 15.30.11. The algebraic version of (4) is More on Algebra, Lemma 15.30.10. $\square$

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