The Stacks project

Lemma 75.26.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_ X)$ be perfect of tor-amplitude in $[a, b]$ for some $a, b \in \mathbf{Z}$. Let $r \geq 0$. Then there exists a locally closed subspace $j : Z \to X$ characterized by the following

1. $H^ a(Lj^*E)$ is a locally free $\mathcal{O}_ Z$-module of rank $r$, and

2. a morphism $f : Y \to X$ factors through $Z$ if and only if for all morphisms $g : Y' \to Y$ the $\mathcal{O}_{Y'}$-module $H^ a(L(f \circ g)^*E)$ is locally free of rank $r$.

Moreover, $j : Z \to X$ is of finite presentation and we have

1. if $f : Y \to X$ factors as $Y \xrightarrow {g} Z \to X$, then $H^ a(Lf^*E) = g^*H^ a(Lj^*E)$,

2. if $\beta _ a(x) \leq r$ for all $x \in |X|$, then $j$ is a closed immersion and given $f : Y \to X$ the following are equivalent

   1. $f : Y \to X$ factors through $Z$,

   2. $H^0(Lf^*E)$ is a locally free $\mathcal{O}_ Y$-module of rank $r$,

   and if $r = 1$ these are also equivalent to

   1. $\mathcal{O}_ Y \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(H^0(Lf^*E), H^0(Lf^*E))$ is injective.