The Stacks project

Lemma 96.15.2. Let $U \subset U'$ be a first order thickening of affine schemes. Let $X'$ be an algebraic space flat over $U'$. Set $X = U \times _{U'} X'$. Let $Z \to U$ be finite locally free of degree $d$. Finally, let $f : Z \to X$ be unramified and a local complete intersection morphism. Then there exists a commutative diagram

\[ \xymatrix{ (Z \subset Z') \ar[rd] \ar[rr]_{(f, f')} & & (X \subset X') \ar[ld] \\ & (U \subset U') } \]

of algebraic spaces over $U'$ such that $Z' \to U'$ is finite locally free of degree $d$ and $Z = U \times _{U'} Z'$.

Proof. By More on Morphisms of Spaces, Lemma 75.48.12 the conormal sheaf $\mathcal{C}_{Z/X}$ of the unramified morphism $Z \to X$ is a finite locally free $\mathcal{O}_ Z$-module and by More on Morphisms of Spaces, Lemma 75.48.13 we have an exact sequence

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

of conormal sheaves. Since $Z$ is affine this sequence is split. Choose a splitting

\[ \mathcal{C}_{Z/X'} = i^*\mathcal{C}_{X/X'} \oplus \mathcal{C}_{Z/X} \]

Let $Z \subset Z''$ be the universal first order thickening of $Z$ over $X'$ (see More on Morphisms of Spaces, Section 75.15). Denote $\mathcal{I} \subset \mathcal{O}_{Z''}$ the quasi-coherent sheaf of ideals corresponding to $Z \subset Z''$. By definition we have $\mathcal{C}_{Z/X'}$ is $\mathcal{I}$ viewed as a sheaf on $Z$. Hence the splitting above determines a splitting

\[ \mathcal{I} = i^*\mathcal{C}_{X/X'} \oplus \mathcal{C}_{Z/X} \]

Let $Z' \subset Z''$ be the closed subscheme cut out by $\mathcal{C}_{Z/X} \subset \mathcal{I}$ viewed as a quasi-coherent sheaf of ideals on $Z''$. It is clear that $Z'$ is a first order thickening of $Z$ and that we obtain a commutative diagram of first order thickenings as in the statement of the lemma.

Since $X' \to U'$ is flat and since $X = U \times _{U'} X'$ we see that $\mathcal{C}_{X/X'}$ is the pullback of $\mathcal{C}_{U/U'}$ to $X$, see More on Morphisms of Spaces, Lemma 75.18.1. Note that by construction $\mathcal{C}_{Z/Z'} = i^*\mathcal{C}_{X/X'}$ hence we conclude that $\mathcal{C}_{Z/Z'}$ is isomorphic to the pullback of $\mathcal{C}_{U/U'}$ to $Z$. Applying More on Morphisms of Spaces, Lemma 75.18.1 once again (or its analogue for schemes, see More on Morphisms, Lemma 37.10.1) we conclude that $Z' \to U'$ is flat and that $Z = U \times _{U'} Z'$. Finally, More on Morphisms, Lemma 37.10.3 shows that $Z' \to U'$ is finite locally free of degree $d$. $\square$


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