The Stacks project

Proof. By More on Morphisms of Spaces, Lemma 76.48.12 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 76.48.13 we have an exact sequence

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

Let $Z \subset Z''$ be the universal first order thickening of $Z$ over $X'$ (see More on Morphisms of Spaces, Section 76.15). 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

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 76.18.1. 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 76.18.1 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 $d$. $\square$