Lemma 37.62.23 (06BA)—The Stacks project

Lemma 37.62.23. Let $Z \to Y \to X$ be formally unramified morphisms of schemes. Assume that $Z \to Y$ is a local complete intersection morphism. The exact sequence

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

of Lemma 37.7.12 is short exact.

Proof. The question is local on $Z$ hence we may assume there exists a factorization $Z \to \mathbf{A}^ n_ Y \to Y$ of the morphism $Z \to Y$ . Then we get a commutative diagram

$\xymatrix{ Z \ar[r]_{i'} \ar@{=}[d] & \mathbf{A}^ n_ Y \ar[r] \ar[d] & \mathbf{A}^ n_ X \ar[d] \\ Z \ar[r]^ i & Y \ar[r] & X }$

As $Z \to Y$ is a local complete intersection morphism, we see that $Z \to \mathbf{A}^ n_ Y$ is a Koszul-regular immersion. Hence by Divisors, Lemma 31.21.6 the sequence

$0 \to (i')^*\mathcal{C}_{\mathbf{A}^ n_ Y/\mathbf{A}^ n_ X} \to \mathcal{C}_{Z/\mathbf{A}^ n_ X} \to \mathcal{C}_{Z/\mathbf{A}^ n_ Y} \to 0$

is exact and locally split. Note that $i^*\mathcal{C}_{Y/X} = (i')^*\mathcal{C}_{\mathbf{A}^ n_ Y/\mathbf{A}^ n_ X}$ by Lemma 37.7.7 and note that the diagram

$\xymatrix{ (i')^*\mathcal{C}_{\mathbf{A}^ n_ Y/\mathbf{A}^ n_ X} \ar[r] & \mathcal{C}_{Z/\mathbf{A}^ n_ X} \\ i^*\mathcal{C}_{Y/X} \ar[u]^{\cong } \ar[r] & \mathcal{C}_{Z/X} \ar[u] }$

is commutative. Hence the lower horizontal arrow is a locally split injection. This proves the lemma. $\square$

The Stacks project

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