The Stacks project

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

  1. $E|_ U \cong H^ i(E|_ U)[-i]$ and $H^ i(E|_ U)$ is a locally free $\mathcal{O}_ U$-module of rank $r$,

  2. 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 15.74.6. 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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