Lemma 36.31.3. Let X be a scheme. Let E \in D(\mathcal{O}_ X) be perfect. Given i, r \in \mathbf{Z}, there exists an open subscheme U \subset X characterized by the following
E|_ U \cong H^ i(E|_ U)[-i] and H^ i(E|_ U) is a locally free \mathcal{O}_ U-module of rank r,
a morphism f : Y \to X factors through U if and only if Lf^*E is isomorphic to a locally free module of rank r placed in degree i.
Proof.
Let \beta _ j : X \to \{ 0, 1, 2, \ldots \} for j \in \mathbf{Z} be the functions of Lemma 36.31.1. Then the set
W = \{ x \in X \mid \beta _ j(x) \leq 0\text{ for all }j \not= i\}
is open in X and its formation commutes with pullback to any Y over X. This follows from the lemma using that apriori in a neighbourhood of any point only a finite number of the \beta _ j are nonzero. Thus we may replace X by W and assume that \beta _ j(x) = 0 for all x \in X and all j \not= i. In this case H^ i(E) is a finite locally free module and E \cong H^ i(E)[-i], see for example More on Algebra, Lemma 15.75.7. Thus X is the disjoint union of the open subschemes where the rank of H^ i(E) is fixed and we win.
\square
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