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 0th 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
\begin{align*} \text{Fit}_0(\varphi ^*\mathcal{F}) & = \text{Fit}_0((W \times _ Y Z \to W)_*\mathcal{O}_{W \times _ Y Z}) \\ & = \text{Fit}_0(\mathcal{O}_ W/\mathcal{I}) \cdot \text{Fit}_0(\tau '_*\mathcal{O}_{Z'}) \\ & = \mathcal{I} \cdot \text{Fit}_0(\tau '_*\mathcal{O}_{Z'}) \end{align*}
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
Comments (2)
Comment #6478 by Takumi Murayama on
Comment #6553 by Johan on