The Stacks project

Lemma 36.37.1. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Let $\mathcal{F}^\bullet $ be a bounded below complex of quasi-coherent $\mathcal{O}_ X$-modules representing a perfect object of $D(\mathcal{O}_ X)$. Then there exists a bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules and a quasi-isomorphism $\mathcal{E}^\bullet \to \mathcal{F}^\bullet $.

Proof. Let $a, b \in \mathbf{Z}$ be integers such that $\mathcal{F}^\bullet $ has tor amplitude in $[a, b]$ and such that $\mathcal{F}^ n = 0$ for $n < a$. The existence of such a pair of integers follows from Cohomology, Lemma 20.46.5 and the fact that $X$ is quasi-compact. If $b < a$, then $\mathcal{F}^\bullet $ is zero in the derived category and the lemma holds. We will prove by induction on $b - a \geq 0$ that there exists a complex $\mathcal{E}^ a \to \ldots \to \mathcal{E}^ b$ with $\mathcal{E}^ i$ finite locally free and a quasi-isomorphism $\mathcal{E}^\bullet \to \mathcal{F}^\bullet $.

The base case is the case $b - a = 0$. In this case $H^ b(\mathcal{F}^\bullet ) = H^ a(\mathcal{F}^\bullet ) = \mathop{\mathrm{Ker}}(\mathcal{F}^ a \to \mathcal{F}^{a + 1})$ is finite locally free. Namely, it is a finitely presented $\mathcal{O}_ X$-module of tor dimension $0$ and hence finite locally free. See Cohomology, Lemmas 20.46.5 and 20.44.9 and Properties, Lemma 28.20.2. Thus we can take $\mathcal{E}^\bullet $ to be $H^ b(\mathcal{F}^\bullet )$ sitting in degree $b$. The rest of the proof is dedicated to the induction step.

Assume $b > a$. Observe that

\[ H^ b(\mathcal{F}^\bullet ) = \mathop{\mathrm{Ker}}(\mathcal{F}^ b \to \mathcal{F}^{b + 1})/ \mathop{\mathrm{Im}}(\mathcal{F}^{b - 1} \to \mathcal{F}^ b) \]

is a finite type quasi-coherent $\mathcal{O}_ X$-module, see Cohomology, Lemmas 20.46.5 and 20.44.9. Then we can find a finite type quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and a map

\[ \mathcal{F} \longrightarrow \mathop{\mathrm{Ker}}(\mathcal{F}^ b \to \mathcal{F}^{b + 1}) \]

such that the composition with the projection onto $H^ b(\mathcal{F}^\bullet )$ is surjective. Namely, we can write $\mathop{\mathrm{Ker}}(\mathcal{F}^ b \to \mathcal{F}^{b + 1})$ as the filtered union of its finite type quasi-coherent submodules by Properties, Lemma 28.22.3 and then one of these will do the job. Next, we choose a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}^ b$ and a surjection $\mathcal{E}^ b \to \mathcal{F}$ using the resolution property of $X$. Consider the map of complexes

\[ \alpha : \mathcal{E}^ b[-b] \to \mathcal{F}^\bullet \]

and its cone $C(\alpha )^\bullet $, see Derived Categories, Definition 13.9.1. We observe that $C(\alpha )^\bullet $ is nonzero only in degrees $\geq a$, has tor amplitude in $[a, b]$ by Cohomology, Lemma 20.45.6, and has $H^ b(C(\alpha )^\bullet ) = 0$ by construction. Thus we actually find that $C(\alpha )^\bullet $ has tor amplitude in $[a, b - 1]$. Hence the induction hypothesis applies to $C(\alpha )^\bullet $ and we find a map of complexes

\[ (\mathcal{E}^ a \to \ldots \to \mathcal{E}^{b - 1}) \longrightarrow C(\alpha )^\bullet \]

with properties as stated in the induction hypothesis. Unwinding the definition of the cone this gives a commutative diagram

\[ \xymatrix{ \ldots \ar[r] & \mathcal{E}^{b - 2} \ar[r] \ar[d] & \mathcal{E}^{b - 1} \ar[r] \ar[d] & 0 \ar[r] \ar[d] & \ldots \\ \ldots \ar[r] & \mathcal{F}^{b - 2} \ar[r] & \mathcal{F}^{b - 1} \oplus \mathcal{E}^ b \ar[r] & \mathcal{F}^ b \ar[r] & \ldots } \]

It is clear that we obtain a map of complexes $(\mathcal{E}^ a \to \ldots \to \mathcal{E}^ b) \to \mathcal{F}^\bullet $. We omit the verification that this map is a quasi-isomorphism. $\square$


Comments (2)

Comment #7259 by Torsten Wedhorn on

Just kind of a typo: One probably should replace "coherent" (appearing twice in the middle of the proof) by "quasi-coherent of finite type".


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