36.37 The resolution property and perfect complexes

In this section we discuss the relationship between perfect complexes and strictly perfect complexes on schemes which have the resolution property.

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 20.49.5 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 20.49.5 and 20.47.9 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 20.49.5 and 20.47.9. 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 20.48.6, 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. $\square$

Lemma 36.37.2. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then every perfect object of $D(\mathcal{O}_ X)$ can be represented by a bounded complex of finite locally free $\mathcal{O}_ X$-modules.

Proof. Let $E$ be a perfect object of $D(\mathcal{O}_ X)$. By Lemma 36.36.10 we see that $X$ has affine diagonal. Hence by Proposition 36.7.5 we can represent $E$ by a complex $\mathcal{F}^\bullet$ of quasi-coherent $\mathcal{O}_ X$-modules. Observe that $E$ is in $D^ b(\mathcal{O}_ X)$ because $X$ is quasi-compact. Hence $\tau _{\geq n}\mathcal{F}^\bullet$ is a bounded below complex of quasi-coherent $\mathcal{O}_ X$-modules which represents $E$ if $n \ll 0$. Thus we may apply Lemma 36.37.1 to conclude. $\square$

Lemma 36.37.3. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Let $\mathcal{E}^\bullet$ and $\mathcal{F}^\bullet$ be finite complexes of finite locally free $\mathcal{O}_ X$-modules. Then any $\alpha \in \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ can be represented by a diagram

$\mathcal{E}^\bullet \leftarrow \mathcal{G}^\bullet \to \mathcal{F}^\bullet$

where $\mathcal{G}^\bullet$ is a bounded complex of finite locally free $\mathcal{O}_ X$-modules and where $\mathcal{G}^\bullet \to \mathcal{E}^\bullet$ is a quasi-isomorphism.

Proof. By Lemma 36.36.10 we see that $X$ has affine diagonal. Hence by Proposition 36.7.5 we can represent $\alpha$ by a diagram

$\mathcal{E}^\bullet \leftarrow \mathcal{H}^\bullet \to \mathcal{F}^\bullet$

where $\mathcal{H}^\bullet$ is a complex of quasi-coherent $\mathcal{O}_ X$-modules and where $\mathcal{H}^\bullet \to \mathcal{E}^\bullet$ is a quasi-isomorphism. For $n \ll 0$ the maps $\mathcal{H}^\bullet \to \mathcal{E}^\bullet$ and $\mathcal{H}^\bullet \to \mathcal{F}^\bullet$ factor through the quasi-isomorphism $\mathcal{H}^\bullet \to \tau _{\geq n}\mathcal{H}^\bullet$ simply because $\mathcal{E}^\bullet$ and $\mathcal{F}^\bullet$ are bounded complexes. Thus we may replace $\mathcal{H}^\bullet$ by $\tau _{\geq n}\mathcal{H}^\bullet$ and assume that $\mathcal{H}^\bullet$ is bounded below. Then we may apply Lemma 36.37.1 to conclude. $\square$

Lemma 36.37.4. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Let $\mathcal{E}^\bullet$ and $\mathcal{F}^\bullet$ be finite complexes of finite locally free $\mathcal{O}_ X$-modules. Let $\alpha ^\bullet , \beta ^\bullet :\mathcal{E}^\bullet \to \mathcal{F}^\bullet$ be two maps of complexes defining the same map in $D(\mathcal{O}_ X)$. Then there exists a quasi-isomorphism $\gamma ^\bullet : \mathcal{G}^\bullet \to \mathcal{E}^\bullet$ where $\mathcal{G}^\bullet$ is a bounded complex of finite locally free $\mathcal{O}_ X$-modules such that $\alpha ^\bullet \circ \gamma ^\bullet$ and $\beta ^\bullet \circ \gamma ^\bullet$ are homotopic maps of complexes.

Proof. By Lemma 36.36.10 we see that $X$ has affine diagonal. Hence by Proposition 36.7.5 (and the definition of the derived category) there exists a quasi-isomorphism $\gamma ^\bullet : \mathcal{G}^\bullet \to \mathcal{E}^\bullet$ where $\mathcal{G}^\bullet$ is a complex of quasi-coherent $\mathcal{O}_ X$-modules such that $\alpha ^\bullet \circ \gamma ^\bullet$ and $\beta ^\bullet \circ \gamma ^\bullet$ are homotopic maps of complexes. Choose a homotopy $h^ i : \mathcal{G}^ i \to \mathcal{F}^{i - 1}$ witnessing this fact. Choose $n \ll 0$. Then the map $\gamma ^\bullet$ factors canonically over the quotient map $\mathcal{G}^\bullet \to \tau _{\geq n}\mathcal{G}^\bullet$ as $\mathcal{E}^\bullet$ is bounded below. For the exact same reason the maps $h^ i$ will factor over the surjections $\mathcal{G}^ i \to (\tau _{\geq n}\mathcal{G})^ i$. Hence we see that we may replace $\mathcal{G}^\bullet$ by $\tau _{\geq n}\mathcal{G}^\bullet$. Then we may apply Lemma 36.37.1 to conclude. $\square$

Proposition 36.37.5. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Denote

1. $\mathcal{A}$ the additive category of finite locally free $\mathcal{O}_ X$-modules,

2. $K^ b(\mathcal{A})$ the homotopy category of bounded complexes in $\mathcal{A}$, see Derived Categories, Section 13.8, and

3. $D_{perf}(\mathcal{O}_ X)$ the strictly full, saturated, triangulated subcategory of $D(\mathcal{O}_ X)$ consisting of perfect objects.

With this notation the obvious functor

$K^ b(\mathcal{A}) \longrightarrow D_{perf}(\mathcal{O}_ X)$

is an exact functor of trianglated categories which factors through an equivalence $S^{-1}K^ b(\mathcal{A}) \to D_{perf}(\mathcal{O}_ X)$ of triangulated categories where $S$ is the saturated multiplicative system of quasi-isomorphisms in $K^ b(\mathcal{A})$.

Proof. If you can parse the statement of the proposition, then please skip this first paragraph. For some of the definitions used, please see Derived Categories, Definition 13.3.4 (triangulated subcategory), Derived Categories, Definition 13.6.1 (saturated triangulated subcategory), Derived Categories, Definition 13.5.1 (multiplicative system compatible with the triangulated structure), and Categories, Definition 4.27.20 (saturated multiplicative system). Observe that $D_{perf}(\mathcal{O}_ X)$ is a saturated triangulated subcategory of $D(\mathcal{O}_ X)$ by Cohomology, Lemmas 20.49.7 and 20.49.9. Also, note that $K^ b(\mathcal{A})$ is a triangulated category, see Derived Categories, Lemma 13.10.5.

It is clear that the functor sends distinguished triangles to distinguished triangles, i.e., is exact. Then $S$ is a saturated multiplicative system compatible with the triangulated structure on $K^ b(\mathcal{A})$ by Derived Categories, Lemma 13.5.4. Hence the localization $S^{-1}K^ b(\mathcal{A})$ exists and is a triangulated category by Derived Categories, Proposition 13.5.6. We get an exact factorization $S^{-1}K^ b(\mathcal{A}) \to D_{perf}(\mathcal{O}_ X)$ by Derived Categories, Lemma 13.5.7. By Lemmas 36.37.2, 36.37.3, and 36.37.4 this functor is an equivalence. Then finally the functor $S^{-1}K^ b(\mathcal{A}) \to D_{perf}(\mathcal{O}_ X)$ is an equivalence of triangulated categories (in the sense that distinguished triangles correspond) by Derived Categories, Lemma 13.4.18. $\square$

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