The Stacks project

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 15.64.2 we see that $C^\bullet _{f_ j}$ is $(n - 1)$-pseudo-coherent. By More on Algebra, Lemma 15.64.3 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 20.49.5. 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 20.49.5 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 20.48.2). 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 20.46.9. Then $\alpha $ determines an element $s$ of $H^0(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, E'))$, see Cohomology, Lemma 20.42.1. 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 20.42.1. $\square$

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 [Neeman-Grothendieck].

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 20.45.1 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.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08EC. Beware of the difference between the letter 'O' and the digit '0'.