The Stacks project

Proof. Choose a factorization $W \to Y \to X$ where $j : W \to Y$ is a quasi-compact open immersion and $\pi : Y \to X$ is a finite morphism of finite presentation (Lemma 76.34.4). Let $V = j(W) \cup \pi ^{-1}(U) \subset Y$. Note that $\mathcal{I}$ on $W \cong j(W)$ and $\mathcal{O}_{\pi ^{-1}(U)}$ glue to a finite type quasi-coherent sheaf of ideals $\mathcal{I}_1 \subset \mathcal{O}_ V$. By Limits of Spaces, Lemma 70.9.8 there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I}_2 \subset \mathcal{O}_ Y$ such that $\mathcal{I}_2|_ V = \mathcal{I}_1$. In other words, $\mathcal{I}_2 \subset \mathcal{O}_ Y$ is a finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I}_2)$ is disjoint from $\pi ^{-1}(U)$ and $j^{-1}\mathcal{I}_2 = \mathcal{I}$. Denote $i : Z \to Y$ the corresponding closed immersion which is of finite presentation (Morphisms of Spaces, Lemma 67.28.12). In particular the composition $\tau = \pi \circ i : Z \to X$ is finite and of finite presentation (Morphisms of Spaces, Lemmas 67.28.2 and 67.45.4).

Let $\mathcal{F} = \tau _*\mathcal{O}_ Z$ which we think of as a quasi-coherent $\mathcal{O}_ X$-module. By Descent on Spaces, Lemma 74.6.7 we see that $\mathcal{F}$ is a finitely presented $\mathcal{O}_ X$-module. Let $\mathcal{J} = \text{Fit}_0(\mathcal{F})$. (Insert reference to fitting modules on ringed topoi here.) This is a finite type quasi-coherent sheaf of ideals on $X$ (as $\mathcal{F}$ is of finite presentation, see More on Algebra, Lemma 15.8.4). Part (1) of the lemma holds because $|\tau |(|Z|) \cap |U| = \emptyset $ by our choice of $\mathcal{I}_2$ and because the $0$th Fitting ideal of the trivial module equals the structure sheaf. To prove (2) note that $\varphi ^{-1}(\mathcal{J})\mathcal{O}_ W = \text{Fit}_0(\varphi ^*\mathcal{F})$ because taking Fitting ideals commutes with base change. On the other hand, as $\varphi : W \to X$ is separated and étale we see that $(1, j) : W \to W \times _ X Y$ is an open and closed immersion. Hence $W \times _ Y Z = V(\mathcal{I}) \amalg Z'$ for some finite and finitely presented morphism of algebraic spaces $\tau ' : Z' \to W$. Thus we see that

the second equality by More on Algebra, Lemma 15.8.4 translated in sheaves on ringed topoi. Setting $\mathcal{I}' = \text{Fit}_0(\tau '_*\mathcal{O}_{Z'})$ finishes the proof of the lemma. $\square$