The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 21.44.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a \in \mathbf{Z}$. The following are equivalent

  1. $E$ has tor-amplitude in $[a, \infty ]$.

  2. $E$ can be represented by a K-flat complex $\mathcal{E}^\bullet $ of flat $\mathcal{O}$-modules with $\mathcal{E}^ i = 0$ for $i \not\in [a, \infty ]$.

Moreover, we can choose $\mathcal{E}^\bullet $ such that any pullback by a morphism of ringed sites is a K-flat complex with flat terms.

Proof. The implication (2) $\Rightarrow $ (1) is immediate. Assume (1) holds. First we choose a K-flat complex $\mathcal{K}^\bullet $ with flat terms representing $E$, see Lemma 21.18.10. For any $\mathcal{O}$-module $\mathcal{M}$ the cohomology of

\[ \mathcal{K}^{n - 1} \otimes _\mathcal {O} \mathcal{M} \to \mathcal{K}^ n \otimes _\mathcal {O} \mathcal{M} \to \mathcal{K}^{n + 1} \otimes _\mathcal {O} \mathcal{M} \]

computes $H^ n(E \otimes _\mathcal {O}^\mathbf {L} \mathcal{M})$. This is always zero for $n < a$. Hence if we apply Lemma 21.44.2 to the complex $\ldots \to \mathcal{K}^{a - 1} \to \mathcal{K}^ a \to \mathcal{K}^{a + 1}$ we conclude that $\mathcal{N} = \mathop{\mathrm{Coker}}(\mathcal{K}^{a - 1} \to \mathcal{K}^ a)$ is a flat $\mathcal{O}$-module. We set

\[ \mathcal{E}^\bullet = \tau _{\geq a}\mathcal{K}^\bullet = (\ldots \to 0 \to \mathcal{N} \to \mathcal{K}^{a + 1} \to \ldots ) \]

The kernel $\mathcal{L}^\bullet $ of $\mathcal{K}^\bullet \to \mathcal{E}^\bullet $ is the complex

\[ \mathcal{L}^\bullet = (\ldots \to \mathcal{K}^{a - 1} \to \mathcal{I} \to 0 \to \ldots ) \]

where $\mathcal{I} \subset \mathcal{K}^ a$ is the image of $\mathcal{K}^{a - 1} \to \mathcal{K}^ a$. Since we have the short exact sequence $0 \to \mathcal{I} \to \mathcal{K}^ a \to \mathcal{N} \to 0$ we see that $\mathcal{I}$ is a flat $\mathcal{O}$-module. Thus $\mathcal{L}^\bullet $ is a bounded above complex of flat modules, hence K-flat by Lemma 21.18.7. It follows that $\mathcal{E}^\bullet $ is K-flat by Lemma 21.18.6.

Proof of the final assertion. Let $f : (\mathcal{C}', \mathcal{O}') \to (\mathcal{C}, \mathcal{O})$ be a morphism of ringed sites. The proof of Lemma 21.19.1 shows that the complex $\mathcal{K}^\bullet $ (as constructed in Lemma 21.18.10) has the property that $f^*\mathcal{K}^\bullet $ is K-flat. The complex $f^*\mathcal{L}^\bullet $ is K-flat as it is a bounded above complex of flat $\mathcal{O}'$-modules. We have a short exact sequence of complexes of $\mathcal{O}'$-modules

\[ 0 \to f^*\mathcal{L}^\bullet \to f^*\mathcal{K}^\bullet \to f^*\mathcal{E}^\bullet \to 0 \]

because the short exact sequence $0 \to \mathcal{I} \to \mathcal{K}^ a \to \mathcal{N} \to 0$ of flat modules pulls back to a short exact sequence. Then we can use the 2-out-of-3 property for K-flat complexes to conclude that $f^*\mathcal{E}^\bullet $ is K-flat. $\square$


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