The Stacks project

75.14 Approximation by perfect complexes

In this section we continue the discussion started in Derived Categories of Schemes, Section 36.14.

Definition 75.14.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider triples $(T, E, m)$ where

  1. $T \subset |X|$ is a closed subset,

  2. $E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$, and

  3. $m \in \mathbf{Z}$.

We say approximation holds for the triple $(T, E, m)$ if there exists a perfect object $P$ of $D(\mathcal{O}_ X)$ supported on $T$ and a map $\alpha : P \to E$ which induces isomorphisms $H^ i(P) \to H^ i(E)$ for $i > m$ and a surjection $H^ m(P) \to H^ m(E)$.

Approximation cannot hold for every triple. Please read the remarks following Derived Categories of Schemes, Definition 36.14.1 to see why.

Definition 75.14.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say approximation by perfect complexes holds on $X$ if for any closed subset $T \subset |X|$ such that the morphism $X \setminus T \to X$ is quasi-compact there exists an integer $r$ such that for every triple $(T, E, m)$ as in Definition 75.14.1 with

  1. $E$ is $(m - r)$-pseudo-coherent, and

  2. $H^ i(E)$ is supported on $T$ for $i \geq m - r$

approximation holds.

Lemma 75.14.3. Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic space over $S$. Let $E$ be a perfect object of $D(\mathcal{O}_ V)$ supported on $j^{-1}(T)$ where $T = |X| \setminus |U|$. Then $Rj_*E$ is a perfect object of $D(\mathcal{O}_ X)$.

Proof. Being perfect is local on $X_{\acute{e}tale}$. Thus it suffices to check that $Rj_*E$ is perfect when restricted to $U$ and $V$. We have $Rj_*E|_ V = E$ by Lemma 75.10.7 which is perfect. We have $Rj_*E|_ U = 0$ because $E|_{V \setminus j^{-1}(T)} = 0$ (use Lemma 75.3.1). $\square$

Lemma 75.14.4. Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Let $T$ be a closed subset of $|X| \setminus |U|$ and let $(T, E, m)$ be a triple as in Definition 75.14.1. If

  1. approximation holds for $(j^{-1}T, E|_ V, m)$, and

  2. the sheaves $H^ i(E)$ for $i \geq m$ are supported on $T$,

then approximation holds for $(T, E, m)$.

Proof. Let $P \to E|_ V$ be an approximation of the triple $(j^{-1}T, E|_ V, m)$ over $V$. Then $Rj_*P$ is a perfect object of $D(\mathcal{O}_ X)$ by Lemma 75.14.3. On the other hand, $Rj_*P = j_!P$ by Lemma 75.10.7. We see that $j_!P$ is supported on $T$ for example by (75.10.0.2). Hence we obtain an approximation $Rj_*P = j_!P \to j_!(E|_ V) \to E$. $\square$

Lemma 75.14.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ which is representable by an affine scheme. Then approximation holds for every triple $(T, E, m)$ as in Definition 75.14.1 such that there exists an integer $r \geq 0$ with

  1. $E$ is $m$-pseudo-coherent,

  2. $H^ i(E)$ is supported on $T$ for $i \geq m - r + 1$,

  3. $X \setminus T$ is the union of $r$ affine opens.

In particular, approximation by perfect complexes holds for affine schemes.

Proof. Let $X_0$ be an affine scheme representing $X$. Let $T_0 \subset X_0$ by the closed subset corresponding to $T$. Let $\epsilon : X_{\acute{e}tale}\to X_{0, Zar}$ be the morphism (75.4.0.1). We may write $E = \epsilon ^*E_0$ for some object $E_0$ of $D_\mathit{QCoh}(\mathcal{O}_{X_0})$, see Lemma 75.4.2. Then $E_0$ is $m$-pseudo-coherent, see Lemma 75.13.2. Comparing stalks of cohomology sheaves (see proof of Lemma 75.4.1) we see that $H^ i(E_0)$ is supported on $T_0$ for $i \geq m - r + 1$. By Derived Categories of Schemes, Lemma 36.14.4 there exists an approximation $P_0 \to E_0$ of $(T_0, E_0, m)$. By Lemma 75.13.5 we see that $P = \epsilon ^*P_0$ is a perfect object of $D(\mathcal{O}_ X)$. Pulling back we obtain an approximation $P = \epsilon ^*P_0 \to \epsilon ^*E_0 = E$ as desired. $\square$

Lemma 75.14.6. Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Assume $U$ quasi-compact, $V$ affine, and $U \times _ X V$ quasi-compact. If approximation by perfect complexes holds on $U$, then approximation by perfect complexes holds on $X$.

Proof. Let $T \subset |X|$ be a closed subset with $X \setminus T \to X$ quasi-compact. Let $r_ U$ be the integer of Definition 75.14.2 adapted to the pair $(U, T \cap |U|)$. Set $T' = T \setminus |U|$. Endow $T'$ with the induced reduced subspace structure. Since $|T'|$ is contained in $|X| \setminus |U|$ we see that $j^{-1}(T') \to T'$ is an isomorphism. Moreover, $V \setminus j^{-1}(T')$ is quasi-compact as it is the fibre product of $U \times _ X V$ with $X \setminus T$ over $X$ and we've assumed $U \times _ X V$ quasi-compact and $X \setminus T \to X$ quasi-compact. Let $r'$ be the number of affines needed to cover $V \setminus j^{-1}(T')$. We claim that $r = \max (r_ U, r')$ works for the pair $(X, T)$.

To see this choose a triple $(T, E, m)$ such that $E$ is $(m - r)$-pseudo-coherent and $H^ i(E)$ is supported on $T$ for $i \geq m - r$. Let $t$ be the largest integer such that $H^ t(E)|_ U$ is nonzero. (Such an integer exists as $U$ is quasi-compact and $E|_ U$ is $(m - r)$-pseudo-coherent.) We will prove that $E$ can be approximated by induction on $t$.

Base case: $t \leq m - r'$. This means that $H^ i(E)$ is supported on $T'$ for $i \geq m - r'$. Hence Lemma 75.14.5 guarantees the existence of an approximation $P \to E|_ V$ of $(T', E|_ V, m)$ on $V$. Applying Lemma 75.14.4 we see that $(T', E, m)$ can be approximated. Such an approximation is also an approximation of $(T, E, m)$.

Induction step. Choose an approximation $P \to E|_ U$ of $(T \cap |U|, E|_ U, m)$. This in particular gives a surjection $H^ t(P) \to H^ t(E|_ U)$. In the rest of the proof we will use the equivalence of Lemma 75.4.2 (and the compatibilities of Remark 75.6.3) for the representable algebraic spaces $V$ and $U \times _ X V$. We will also use the fact that $(m - r)$-pseudo-coherence, resp. perfectness on the Zariski site and ├ętale site agree, see Lemmas 75.13.2 and 75.13.5. Thus we can use the results of Derived Categories of Schemes, Section 36.13 for the open immersion $U \times _ X V \subset V$. In this way Derived Categories of Schemes, Lemma 36.13.9 implies there exists a perfect object $Q$ in $D(\mathcal{O}_ V)$ supported on $j^{-1}(T)$ and an isomorphism $Q|_{U \times _ X V} \to (P \oplus P[1])|_{U \times _ X V}$. By Derived Categories of Schemes, Lemma 36.13.6 we can replace $Q$ by $Q \otimes ^\mathbf {L} I$ and assume that the map

\[ Q|_{U \times _ X V} \longrightarrow (P \oplus P[1])|_{U \times _ X V} \longrightarrow P|_{U \times _ X V} \longrightarrow E|_{U \times _ X V} \]

lifts to $Q \to E|_ V$. By Lemma 75.10.8 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 $T$ and the map $H^ t(R) \to H^ t(E)$ is surjective on restriction to $U$. Choose a distinguished triangle

\[ R \to E \to E' \to R[1] \]

Then $E'$ is $(m - r)$-pseudo-coherent (Cohomology on Sites, Lemma 21.45.4), $H^ i(E')|_ U = 0$ for $i \geq t$, and $H^ i(E')$ is supported on $T$ for $i \geq m - r$. By induction we find an approximation $R' \to E'$ of $(T, E', m)$. Fit the composition $R' \to E' \to R[1]$ into a distinguished triangle $R \to R'' \to R' \to R[1]$ and extend the morphisms $R' \to E'$ and $R[1] \to R[1]$ into a morphism of distinguished triangles

\[ \xymatrix{ R \ar[r] \ar[d] & R'' \ar[d] \ar[r] & R' \ar[d] \ar[r] & R[1] \ar[d] \\ R \ar[r] & E \ar[r] & E' \ar[r] & R[1] } \]

using TR3. Then $R''$ is a perfect complex (Cohomology on Sites, Lemma 21.47.6) supported on $T$. An easy diagram chase shows that $R'' \to E$ is the desired approximation. $\square$

Theorem 75.14.7. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Then approximation by perfect complexes holds on $X$.

Proof. This follows from the induction principle of Lemma 75.9.3 and Lemmas 75.14.6 and 75.14.5. $\square$


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