Lemma 76.44.9. Let $S$ be a scheme. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of algebraic spaces over $S$. Assume that the sequence
\[ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 \]
of Lemma 76.5.6 is exact and locally split.
If $j \circ i$ is a quasi-regular immersion, so is $i$.
If $j \circ i$ is a $H_1$-regular immersion, so is $i$.
If both $j$ and $j \circ i$ are Koszul-regular immersions, so is $i$.
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