The Stacks project

Lemma 76.44.10. Let $S$ be a scheme. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of algebraic spaces over $S$. Assume $X$ is locally Noetherian. The following are equivalent

  1. $i$ and $j$ are Koszul regular immersions,

  2. $i$ and $j \circ i$ are Koszul regular immersions,

  3. $j \circ i$ is a Koszul regular immersion and the conormal sequence

    \[ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 \]

    is exact and locally split.

Proof. Immediate from the case of schemes, see Divisors, Lemma 31.21.9. $\square$

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