The Stacks project

Lemma 76.44.8. Let $S$ be a scheme. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of algebraic spaces over $S$.

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

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

  3. 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. Immediate from the case of schemes, see Divisors, Lemma 31.21.7. $\square$

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