Lemma 71.5.3. Let S be a scheme. Let X be an algebraic space over S. Let \mathcal{F} be a finite type, quasi-coherent \mathcal{O}_ X-module. Let Z_0 \subset X be the closed subspace cut out by \text{Fit}_0(\mathcal{F}). Let Z \subset X be the scheme theoretic support of \mathcal{F}. Then
Z \subset Z_0 \subset X as closed subspaces,
|Z| = |Z_0| = \text{Supp}(\mathcal{F}) as closed subsets of |X|,
there exists a finite type, quasi-coherent \mathcal{O}_{Z_0}-module \mathcal{G}_0 with
(Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}.
Comments (0)