## 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 étale 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. $\square$

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).