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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