The Stacks project

Lemma 36.31.4. Let $X$ be a scheme. 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 subscheme $j : Z \to X$ 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.

Proof. First, let $U \subset X$ be the locally constructible open subscheme where the function $\beta _ a$ of Lemma 36.31.1 has values $\leq r$. Let $f : Y \to X$ be as in (2). Then for any $y \in Y$ we have $\beta _ a(Lf^*E) = r$ hence $y$ maps into $U$ by Lemma 36.31.1. Hence $f$ as in (2) factors through $U$. Thus we may replace $X$ by $U$ and assume that $\beta _ a(x) \in \{ 0, 1, \ldots , r\} $ for all $x \in X$. We will show that in this case there is a closed subscheme $Z \subset X$ cut out by a finite type quasi-coherent ideal characterized by the equivalence of (4) (a), (b) and (4)(c) if $r = 1$ and that (3) holds. This will finish the proof because it will a fortiori show that morphisms as in (2) factor through $Z$.

If $x \in X$ and $\beta _ a(x) < r$, then there is an open neighbourhood of $x$ where $\beta _ a < r$ (Lemma 36.31.1). In this way we see that set theoretically at least $Z$ is a closed subset.

To get a scheme theoretic structure, consider a point $x \in X$ with $\beta _ a(x) = r$. Set $\beta = \beta _{a + 1}(x)$. By More on Algebra, Lemma 15.75.6 there exists an affine open neighbourhood $U$ of $x$ such that $K|_ U$ is represented by a complex

\[ \ldots \to 0 \to \mathcal{O}_ U^{\oplus r} \xrightarrow {(f_{ij})} \mathcal{O}_ U^{\oplus \beta } \to \ldots \to \mathcal{O}_ U^{\oplus \beta _{b - 1}(x)} \to \mathcal{O}_ U^{\oplus \beta _ b(x)} \to 0 \to \ldots \]

(This also uses earlier results to turn the problem into algebra, for example Lemmas 36.3.5 and 36.10.7.) Now, if $g : Y \to U$ is any morphism of schemes such that $g^\sharp (f_{ij})$ is nonzero for some pair $i, j$, then $H^0(Lg^*E)$ is not a locally free $\mathcal{O}_ Y$-module of rank $r$. See More on Algebra, Lemma 15.15.7. Trivially $H^0(Lg^*E)$ is a locally free $\mathcal{O}_ Y$-module if $g^\sharp (f_{ij}) = 0$ for all $i, j$. Thus we see that over $U$ the closed subscheme cut out by all $f_{ij}$ satisfies (3) and we have the equivalence of (4)(a) and (b). The characterization of $Z$ shows that the locally constructed patches glue (details omitted). Finally, if $r = 1$ then (4)(c) is equivalent to (4)(b) because in this case locally $H^0(Lg^*E) \subset \mathcal{O}_ Y$ is the annihilator of the ideal generated by the elements $g^\sharp (f_{ij})$. $\square$

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