## 36.13 Lifting complexes

Let $U \subset X$ be an open subspace of a ringed space and denote $j : U \to X$ the inclusion morphism. The functor $D(\mathcal{O}_ X) \to D(\mathcal{O}_ U)$ is essentially surjective as $Rj_*$ is a right inverse to restriction. In this section we extend this to complexes with quasi-coherent cohomology sheaves, etc.

Lemma 36.13.1. Let $X$ be a scheme and let $j : U \to X$ be a quasi-compact open immersion. The functors

$D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ U) \quad \text{and}\quad D^+_\mathit{QCoh}(\mathcal{O}_ X) \to D^+_\mathit{QCoh}(\mathcal{O}_ U)$

are essentially surjective. If $X$ is quasi-compact, then the functors

$D^-_\mathit{QCoh}(\mathcal{O}_ X) \to D^-_\mathit{QCoh}(\mathcal{O}_ U) \quad \text{and}\quad D^ b_\mathit{QCoh}(\mathcal{O}_ X) \to D^ b_\mathit{QCoh}(\mathcal{O}_ U)$

are essentially surjective.

Proof. The argument preceding the lemma applies for the first case because $Rj_*$ maps $D_\mathit{QCoh}(\mathcal{O}_ U)$ into $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.4.1. It is clear that $Rj_*$ maps $D^+_\mathit{QCoh}(\mathcal{O}_ U)$ into $D^+_\mathit{QCoh}(\mathcal{O}_ X)$ which implies the statement on bounded below complexes. Finally, Lemma 36.4.1 guarantees that $Rj_*$ maps $D^-_\mathit{QCoh}(\mathcal{O}_ U)$ into $D^-_\mathit{QCoh}(\mathcal{O}_ X)$ if $X$ is quasi-compact. Combining these two we obtain the last statement. $\square$

Lemma 36.13.2. Let $X$ be a Noetherian scheme and let $j : U \to X$ be an open immersion. The functor $D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ U)$ is essentially surjective.

Proof. Let $K$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$. By Proposition 36.11.2 we can represent $K$ by a bounded complex $\mathcal{F}^\bullet$ of coherent $\mathcal{O}_ U$-modules. Say $\mathcal{F}^ i = 0$ for $i \not\in [a, b]$ for some $a \leq b$. Since $j$ is quasi-compact and separated, the terms of the bounded complex $j_*\mathcal{F}^\bullet$ are quasi-coherent modules on $X$, see Schemes, Lemma 26.24.1. We inductively pick a coherent submodule $\mathcal{G}^ i \subset j_*\mathcal{F}^ i$ as follows. For $i = a$ we pick any coherent submodule $\mathcal{G}^ a \subset j_*\mathcal{F}^ a$ whose restriction to $U$ is $\mathcal{F}^ a$. This is possible by Properties, Lemma 28.22.2. For $i > a$ we first pick any coherent submodule $\mathcal{H}^ i \subset j_*\mathcal{F}^ i$ whose restriction to $U$ is $\mathcal{F}^ i$ and then we set $\mathcal{G}^ i = \mathop{\mathrm{Im}}(\mathcal{H}^ i \oplus \mathcal{G}^{i - 1} \to j_*\mathcal{F}^ i)$. It is clear that $\mathcal{G}^\bullet \subset j_*\mathcal{F}^\bullet$ is a bounded complex of coherent $\mathcal{O}_ X$-modules whose restriction to $U$ is $\mathcal{F}^\bullet$ as desired. $\square$

Lemma 36.13.3. Let $X$ be an affine scheme and let $U \subset X$ be a quasi-compact open subscheme. For any pseudo-coherent object $E$ of $D(\mathcal{O}_ U)$ there exists a bounded above complex of finite free $\mathcal{O}_ X$-modules whose restriction to $U$ is isomorphic to $E$.

Proof. By Lemma 36.10.1 we see that $E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ U)$. By Lemma 36.13.1 we may assume $E = E'|U$ for some object $E'$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Write $X = \mathop{\mathrm{Spec}}(A)$. By Lemma 36.3.5 we can find a complex $M^\bullet$ of $A$-modules whose associated complex of $\mathcal{O}_ X$-modules is a representative of $E'$.

Choose $f_1, \ldots , f_ r \in A$ such that $U = D(f_1) \cup \ldots \cup D(f_ r)$. By Lemma 36.10.2 the complexes $M^\bullet _{f_ j}$ are pseudo-coherent complexes of $A_{f_ j}$-modules. Let $n$ be an integer. Assume we have a map of complexes $\alpha : F^\bullet \to M^\bullet$ where $F^\bullet$ is bounded above, $F^ i = 0$ for $i < n$, each $F^ i$ is a finite free $R$-module, such that

$H^ i(\alpha _{f_ j}) : H^ i(F^\bullet _{f_ j}) \to H^ i(M^\bullet _{f_ j})$

is an isomorphism for $i > n$ and surjective for $i = n$. Picture

$\xymatrix{ & F^ n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \\ M^{n-1} \ar[r] & M^ n \ar[r] & M^{n + 1} \ar[r] & \ldots }$

Since each $M^\bullet _{f_ j}$ has vanishing cohomology in large degrees we can find such a map for $n \gg 0$. By induction on $n$ we are going to extend this to a map of complexes $F^\bullet \to M^\bullet$ such that $H^ i(\alpha _{f_ j})$ is an isomorphism for all $i$. The lemma will follow by taking $\widetilde{F^\bullet }$.

The induction step will be to extend the diagram above by adding $F^{n - 1}$. Let $C^\bullet$ be the cone on $\alpha$ (Derived Categories, Definition 13.9.1). The long exact sequence of cohomology shows that $H^ i(C^\bullet _{f_ j}) = 0$ for $i \geq n$. By More on Algebra, Lemma 15.63.2 we see that $C^\bullet _{f_ j}$ is $(n - 1)$-pseudo-coherent. By More on Algebra, Lemma 15.63.3 we see that $H^{-1}(C^\bullet _{f_ j})$ is a finite $A_{f_ j}$-module. Choose a finite free $A$-module $F^{n - 1}$ and an $A$-module $\beta : F^{n - 1} \to C^{-1}$ such that the composition $F^{n - 1} \to C^{n - 1} \to C^ n$ is zero and such that $F^{n - 1}_{f_ j}$ surjects onto $H^{n - 1}(C^\bullet _{f_ j})$. (Some details omitted; hint: clear denominators.) Since $C^{n - 1} = M^{n - 1} \oplus F^ n$ we can write $\beta = (\alpha ^{n - 1}, -d^{n - 1})$. The vanishing of the composition $F^{n - 1} \to C^{n - 1} \to C^ n$ implies these maps fit into a morphism of complexes

$\xymatrix{ & F^{n - 1} \ar[d]^{\alpha ^{n - 1}} \ar[r]_{d^{n - 1}} & F^ n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \\ \ldots \ar[r] & M^{n - 1} \ar[r] & M^ n \ar[r] & M^{n + 1} \ar[r] & \ldots }$

Moreover, these maps define a morphism of distinguished triangles

$\xymatrix{ (F^ n \to \ldots ) \ar[r] \ar[d] & (F^{n-1} \to \ldots ) \ar[r] \ar[d] & F^{n-1} \ar[r] \ar[d]_\beta & (F^ n \to \ldots )[1] \ar[d] \\ (F^ n \to \ldots ) \ar[r] & M^\bullet \ar[r] & C^\bullet \ar[r] & (F^ n \to \ldots )[1] }$

Hence our choice of $\beta$ implies that the map of complexes $(F^{-1} \to \ldots ) \to M^\bullet$ induces an isomorphism on cohomology localized at $f_ j$ in degrees $\geq n$ and a surjection in degree $-1$. This finishes the proof of the lemma. $\square$

Lemma 36.13.4. Let $X$ be a quasi-compact and quasi-separated scheme. Let $E \in D^ b_\mathit{QCoh}(\mathcal{O}_ X)$. There exists an integer $n_0 > 0$ such that $\mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ X)}(\mathcal{E}, E) = 0$ for every finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ and every $n \geq n_0$.

Proof. Recall that $\mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ X)}(\mathcal{E}, E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}, E[n])$. We have Mayer-Vietoris for morphisms in the derived category, see Cohomology, Lemma 20.33.3. Thus if $X = U \cup V$ and the result of the lemma holds for $E|_ U$, $E|_ V$, and $E|_{U \cap V}$ for some bound $n_0$, then the result holds for $E$ with bound $n_0 + 1$. Thus it suffices to prove the lemma when $X$ is affine, see Cohomology of Schemes, Lemma 30.4.1.

Assume $X = \mathop{\mathrm{Spec}}(A)$ is affine. Choose a complex of $A$-modules $M^\bullet$ whose associated complex of quasi-coherent modules represents $E$, see Lemma 36.3.5. Write $\mathcal{E} = \widetilde{P}$ for some $A$-module $P$. Since $\mathcal{E}$ is finite locally free, we see that $P$ is a finite projective $A$-module. We have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}, E[n]) & = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(P, M^\bullet [n]) \\ & = \mathop{\mathrm{Hom}}\nolimits _{K(A)}(P, M^\bullet [n]) \\ & = \mathop{\mathrm{Hom}}\nolimits _ A(P, H^ n(M^\bullet )) \end{align*}

The first equality by Lemma 36.3.5, the second equality by Derived Categories, Lemma 13.19.8, and the final equality because $\mathop{\mathrm{Hom}}\nolimits _ A(P, -)$ is an exact functor. As $E$ and hence $M^\bullet$ is bounded we get zero for all sufficiently large $n$. $\square$

Lemma 36.13.5. Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. For every perfect object $E$ of $D(\mathcal{O}_ U)$ there exists an integer $r$ and a finite locally free sheaf $\mathcal{F}$ on $U$ such that $\mathcal{F}[-r] \oplus E$ is the restriction of a perfect object of $D(\mathcal{O}_ X)$.

Proof. Say $X = \mathop{\mathrm{Spec}}(A)$. Recall that a perfect complex is pseudo-coherent, see Cohomology, Lemma 20.46.5. By Lemma 36.13.3 we can find a bounded above complex $\mathcal{F}^\bullet$ of finite free $A$-modules such that $E$ is isomorphic to $\mathcal{F}^\bullet |_ U$ in $D(\mathcal{O}_ U)$. By Cohomology, Lemma 20.46.5 and since $U$ is quasi-compact, we see that $E$ has finite tor dimension, say $E$ has tor amplitude in $[a, b]$. Pick $r < a$ and set

$\mathcal{F} = \mathop{\mathrm{Ker}}(\mathcal{F}^{r} \to \mathcal{F}^{r + 1}) = \mathop{\mathrm{Im}}(\mathcal{F}^{r - 1} \to \mathcal{F}^ r).$

Since $E$ has tor amplitude in $[a, b]$ we see that $\mathcal{F}|_ U$ is flat (Cohomology, Lemma 20.45.2). Hence $\mathcal{F}|_ U$ is flat and of finite presentation, thus finite locally free (Properties, Lemma 28.20.2). It follows that

$(\mathcal{F} \to \mathcal{F}^ r \to \mathcal{F}^{r + 1} \to \ldots )|_ U$

is a strictly perfect complex on $U$ representing $E$. We obtain a distinguished triangle

$\mathcal{F}|_ U[- r - 1] \to E \to (\mathcal{F}^ r \to \mathcal{F}^{r + 1} \to \ldots )|_ U \to \mathcal{F}|_ U[- r]$

Note that $(\mathcal{F}^ r \to \mathcal{F}^{r + 1} \to \ldots )$ is a perfect complex on $X$. To finish the proof it suffices to pick $r$ such that the map $\mathcal{F}|_ U[- r - 1] \to E$ is zero in $D(\mathcal{O}_ U)$, see Derived Categories, Lemma 13.4.11. By Lemma 36.13.4 this holds if $r \ll 0$. $\square$

Lemma 36.13.6. Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. Let $E, E'$ be objects of $D_\mathit{QCoh}(\mathcal{O}_ X)$ with $E$ perfect. For every map $\alpha : E|_ U \to E'|_ U$ there exist maps

$E \xleftarrow {\beta } E_1 \xrightarrow {\gamma } E'$

of perfect complexes on $X$ such that $\beta : E_1 \to E$ restricts to an isomorphism on $U$ and such that $\alpha = \gamma |_ U \circ \beta |_ U^{-1}$. Moreover we can assume $E_1 = E \otimes _{\mathcal{O}_ X}^\mathbf {L} I$ for some perfect complex $I$ on $X$.

Proof. Write $X = \mathop{\mathrm{Spec}}(A)$. Write $U = D(f_1) \cup \ldots \cup D(f_ r)$. Choose finite complex of finite projective $A$-modules $M^\bullet$ representing $E$ (Lemma 36.10.7). Choose a complex of $A$-modules $(M')^\bullet$ representing $E'$ (Lemma 36.3.5). In this case the complex $H^\bullet = \mathop{\mathrm{Hom}}\nolimits _ A(M^\bullet , (M')^\bullet )$ is a complex of $A$-modules whose associated complex of quasi-coherent $\mathcal{O}_ X$-modules represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, E')$, see Cohomology, Lemma 20.43.9. Then $\alpha$ determines an element $s$ of $H^0(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, E'))$, see Cohomology, Lemma 20.39.1. There exists an $e$ and a map

$\xi : I^\bullet (f_1^ e, \ldots , f_ r^ e) \to \mathop{\mathrm{Hom}}\nolimits _ A(M^\bullet , (M')^\bullet )$

corresponding to $s$, see Proposition 36.9.5. Letting $E_1$ be the object corresponding to complex of quasi-coherent $\mathcal{O}_ X$-modules associated to

$\text{Tot}(I^\bullet (f_1^ e, \ldots , f_ r^ e) \otimes _ A M^\bullet )$

we obtain $E_1 \to E$ using the canonical map $I^\bullet (f_1^ e, \ldots , f_ r^ e) \to A$ and $E_1 \to E'$ using $\xi$ and Cohomology, Lemma 20.39.1. $\square$

Lemma 36.13.7. Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. For every perfect object $F$ of $D(\mathcal{O}_ U)$ the object $F \oplus F[1]$ is the restriction of a perfect object of $D(\mathcal{O}_ X)$.

Proof. By Lemma 36.13.5 we can find a perfect object $E$ of $D(\mathcal{O}_ X)$ such that $E|_ U = \mathcal{F}[r] \oplus F$ for some finite locally free $\mathcal{O}_ U$-module $\mathcal{F}$. By Lemma 36.13.6 we can find a morphism of perfect complexes $\alpha : E_1 \to E$ such that $(E_1)|_ U \cong E|_ U$ and such that $\alpha |_ U$ is the map

$\left( \begin{matrix} \text{id}_{\mathcal{F}[r]} & 0 \\ 0 & 0 \end{matrix} \right) : \mathcal{F}[r] \oplus F \to \mathcal{F}[r] \oplus F$

Then the cone on $\alpha$ is a solution. $\square$

Lemma 36.13.8. Let $X$ be a quasi-compact and quasi-separated scheme. Let $f \in \Gamma (X, \mathcal{O}_ X)$. For any morphism $\alpha : E \to E'$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ such that

1. $E$ is perfect, and

2. $E'$ is supported on $T = V(f)$

there exists an $n \geq 0$ such that $f^ n \alpha = 0$.

Proof. We have Mayer-Vietoris for morphisms in the derived category, see Cohomology, Lemma 20.33.3. Thus if $X = U \cup V$ and the result of the lemma holds for $f|_ U$, $f|_ V$, and $f|_{U \cap V}$, then the result holds for $f$. Thus it suffices to prove the lemma when $X$ is affine, see Cohomology of Schemes, Lemma 30.4.1.

Let $X = \mathop{\mathrm{Spec}}(A)$. Then $f \in A$. We will use the equivalence $D(A) = D_\mathit{QCoh}(X)$ of Lemma 36.3.5 without further mention. Represent $E$ by a finite complex of finite projective $A$-modules $P^\bullet$. This is possible by Lemma 36.10.7. Let $t$ be the largest integer such that $P^ t$ is nonzero. The distinguished triangle

$P^ t[-t] \to P^\bullet \to \sigma _{\leq t - 1}P^\bullet \to P^ t[-t + 1]$

shows that by induction on the length of the complex $P^\bullet$ we can reduce to the case where $P^\bullet$ has a single nonzero term. This and the shift functor reduces us to the case where $P^\bullet$ consists of a single finite projective $A$-module $P$ in degree $0$. Represent $E'$ by a complex $M^\bullet$ of $A$-modules. Then $\alpha$ corresponds to a map $P \to H^0(M^\bullet )$. Since the module $H^0(M^\bullet )$ is supported on $V(f)$ by assumption (2) we see that every element of $H^0(M^\bullet )$ is annihilated by a power of $f$. Since $P$ is a finite $A$-module the map $f^ n\alpha : P \to H^0(M^\bullet )$ is zero for some $n$ as desired. $\square$

Lemma 36.13.9. Let $X$ be an affine scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. Let $U \subset X$ be a quasi-compact open. For every perfect object $F$ of $D(\mathcal{O}_ U)$ supported on $T \cap U$ the object $F \oplus F[1]$ is the restriction of a perfect object $E$ of $D(\mathcal{O}_ X)$ supported in $T$.

Proof. Say $T = V(g_1, \ldots , g_ s)$. After replacing $g_ j$ by a power we may assume multiplication by $g_ j$ is zero on $F$, see Lemma 36.13.8. Choose $E$ as in Lemma 36.13.7. Note that $g_ j : E \to E$ restricts to zero on $U$. Choose a distinguished triangle

$E \xrightarrow {g_1} E \to C_1 \to E[1]$

By Derived Categories, Lemma 13.4.11 the object $C_1$ restricts to $F \oplus F[1] \oplus F[1] \oplus F[2]$ on $U$. Moreover, $g_1 : C_1 \to C_1$ has square zero by Derived Categories, Lemma 13.4.5. Namely, the diagram

$\xymatrix{ E \ar[r] \ar[d]_0 & C_1 \ar[d]_{g_1} \ar[r] & E[1] \ar[d]_0 \\ E \ar[r] & C_1 \ar[r] & E[1] }$

is commutative since the compositions $E \xrightarrow {g_1} E \to C_1$ and $C_1 \to E[1] \xrightarrow {g_1} E[1]$ are zero. Continuing, setting $C_{i + 1}$ equal to the cone of the map $g_ i : C_ i \to C_ i$ we obtain a perfect complex $C_ s$ on $X$ supported on $T$ whose restriction to $U$ gives

$F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s]$

Choose morphisms of perfect complexes $\beta : C' \to C_ s$ and $\gamma : C' \to C_ s$ as in Lemma 36.13.6 such that $\beta |_ U$ is an isomorphism and such that $\gamma |_ U \circ \beta |_ U^{-1}$ is the morphism

$F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s] \to F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s]$

which is the identity on all summands except for $F$ where it is zero. By Lemma 36.13.6 we also have $C' = C_ s \otimes ^\mathbf {L} I$ for some perfect complex $I$ on $X$. Hence the nullity of $g_ j^2\text{id}_{C_ s}$ implies the same thing for $C'$. Thus $C'$ is supported on $T$ as well. Then $\text{Cone}(\gamma )$ is a solution. $\square$

A special case of the following lemma can be found in .

Lemma 36.13.10. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open. Let $T \subset X$ be a closed subset with $X \setminus T$ retro-compact in $X$. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $\alpha : P \to E|_ U$ be a map where $P$ is a perfect object of $D(\mathcal{O}_ U)$ supported on $T \cap U$. Then there exists a map $\beta : R \to E$ where $R$ is a perfect object of $D(\mathcal{O}_ X)$ supported on $T$ such that $P$ is a direct summand of $R|_ U$ in $D(\mathcal{O}_ U)$ compatible $\alpha$ and $\beta |_ U$.

Proof. Since $X$ is quasi-compact there exists an integer $m$ such that $X = U \cup V_1 \cup \ldots \cup V_ m$ for some affine opens $V_ j$ of $X$. Arguing by induction on $m$ we see that we may assume $m = 1$. In other words, we may assume that $X = U \cup V$ with $V$ affine. By Lemma 36.13.9 we can choose a perfect object $Q$ in $D(\mathcal{O}_ V)$ supported on $T \cap V$ and an isomorphism $Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}$. By Lemma 36.13.6 we can replace $Q$ by $Q \otimes ^\mathbf {L} I$ (still supported on $T \cap V$) and assume that the map

$Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V} \longrightarrow P|_{U \cap V} \longrightarrow E|_{U \cap V}$

lifts to $Q \to E|_ V$. By Cohomology, Lemma 20.42.1 we find an morphism $a : R \to E$ of $D(\mathcal{O}_ X)$ such that $a|_ U$ is isomorphic to $P \oplus P[1] \to E|_ U$ and $a|_ V$ isomorphic to $Q \to E|_ V$. Thus $R$ is perfect and supported on $T$ as desired. $\square$

Remark 36.13.11. The proof of Lemma 36.13.10 shows that

$R|_ U = P \oplus P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_ m}[m]$

for some $m \geq 0$ and $n_ j \geq 0$. Thus the highest degree cohomology sheaf of $R|_ U$ equals that of $P$. By repeating the construction for the map $P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_ m}[m] \to R|_ U$, taking cones, and using induction we can achieve equality of cohomology sheaves of $R|_ U$ and $P$ above any given degree.

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