Lemma 31.22.9. Let

be a commutative diagram of morphisms of schemes. Assume $X \to S$ smooth, and $i$, $j$ immersions. If $j$ is a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) immersion, then so is $i$.

Lemma 31.22.9. Let

\[ \xymatrix{ Y \ar[rd]_ j \ar[rr]_ i & & X \ar[ld] \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume $X \to S$ smooth, and $i$, $j$ immersions. If $j$ is a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) immersion, then so is $i$.

**Proof.**
We can write $i$ as the composition

\[ Y \to Y \times _ S X \to X \]

By Lemma 31.22.8 the first arrow is a regular immersion. The second arrow is a flat base change of $Y \to S$, hence is a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) immersion, see Lemma 31.21.4. We conclude by an application of Lemma 31.21.7. $\square$

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