Definition 36.14.1. Let $X$ be a scheme. Consider triples $(T, E, m)$ where
$T \subset X$ is a closed subset,
$E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$, and
$m \in \mathbf{Z}$.
36.14 Approximation by perfect complexes
In this section we discuss the observation, due to Neeman and Lipman, that a pseudo-coherent complex can be “approximated” by perfect complexes.
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. Namely, it is clear that if approximation holds for the triple $(T, E, m)$, then
$E$ is $m$-pseudo-coherent, see Cohomology, Definition 20.47.1, and
the cohomology sheaves $H^ i(E)$ are supported on $T$ for $i \geq m$.
Moreover, the “support” of a perfect complex is a closed subscheme whose complement is retrocompact in $X$ (details omitted). Hence we cannot expect approximation to hold without this assumption on $T$. This partly explains the conditions in the following definition.
Definition 36.14.2. Let $X$ be a scheme. We say approximation by perfect complexes holds on $X$ if for any closed subset $T \subset X$ with $X \setminus T$ retro-compact in $X$ there exists an integer $r$ such that for every triple $(T, E, m)$ as in Definition 36.14.1 with
$E$ is $(m - r)$-pseudo-coherent, and
$H^ i(E)$ is supported on $T$ for $i \geq m - r$
approximation holds.
We will prove that approximation by perfect complexes holds for quasi-compact and quasi-separated schemes. It seems that the second condition is necessary for our method of proof. It is possible that the first condition may be weakened to “$E$ is $m$-pseudo-coherent” by carefuly analyzing the arguments below.
Lemma 36.14.3. Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. Let $(T, E, m)$ be a triple as in Definition 36.14.1. If
$T \subset U$,
approximation holds for $(T, E|_ U, m)$, and
the sheaves $H^ i(E)$ for $i \geq m$ are supported on $T$,
then approximation holds for $(T, E, m)$.
Proof. Let $j : U \to X$ be the inclusion morphism. If $P \to E|_ U$ is an approximation of the triple $(T, E|_ U, m)$ over $U$, then $j_!P = Rj_*P \to j_!(E|_ U) \to E$ is an approximation of $(T, E, m)$ over $X$. See Cohomology, Lemmas 20.33.6 and 20.49.10. $\square$
Lemma 36.14.4. Let $X$ be an affine scheme. Then approximation holds for every triple $(T, E, m)$ as in Definition 36.14.1 such that there exists an integer $r \geq 0$ with
$E$ is $m$-pseudo-coherent,
$H^ i(E)$ is supported on $T$ for $i \geq m - r + 1$,
$X \setminus T$ is the union of $r$ affine opens.
In particular, approximation by perfect complexes holds for affine schemes.
Proof. Say $X = \mathop{\mathrm{Spec}}(A)$. Write $T = V(f_1, \ldots , f_ r)$. (The case $r = 0$, i.e., $T = X$ follows immediately from Lemma 36.10.2 and the definitions.) Let $(T, E, m)$ be a triple as in the lemma. Let $t$ be the largest integer such that $H^ t(E)$ is nonzero. We will proceed by induction on $t$. The base case is $t < m$; in this case the result is trivial. Now suppose that $t \geq m$. By Cohomology, Lemma 20.47.9 the sheaf $H^ t(E)$ is of finite type. Since it is quasi-coherent it is generated by finitely many sections (Properties, Lemma 28.16.1). For every $s \in \Gamma (X, H^ t(E)) = H^ t(X, E)$ (see proof of Lemma 36.3.5) we can find an $e > 0$ and a morphism $K_ e[-t] \to E$ such that $s$ is in the image of $H^0(K_ e) = H^ t(K_ e[-t]) \to H^ t(E)$, see Lemma 36.9.6. Taking a finite direct sum of these maps we obtain a map $P \to E$ where $P$ is a perfect complex supported on $T$, where $H^ i(P) = 0$ for $i > t$, and where $H^ t(P) \to E$ is surjective. Choose a distinguished triangle
\[ P \to E \to E' \to P[1] \]
Then $E'$ is $m$-pseudo-coherent (Cohomology, Lemma 20.47.4), $H^ i(E') = 0$ for $i \geq t$, and $H^ i(E')$ is supported on $T$ for $i \geq m - r + 1$. By induction we find an approximation $P' \to E'$ of $(T, E', m)$. Fit the composition $P' \to E' \to P[1]$ into a distinguished triangle $P \to P'' \to P' \to P[1]$ and extend the morphisms $P' \to E'$ and $P[1] \to P[1]$ into a morphism of distinguished triangles
\[ \xymatrix{ P \ar[r] \ar[d] & P'' \ar[d] \ar[r] & P' \ar[d] \ar[r] & P[1] \ar[d] \\ P \ar[r] & E \ar[r] & E' \ar[r] & P[1] } \]
using TR3. Then $P''$ is a perfect complex (Cohomology, Lemma 20.49.7) supported on $T$. An easy diagram chase shows that $P'' \to E$ is the desired approximation. $\square$
Lemma 36.14.5. Let $X$ be a scheme. Let $X = U \cup V$ be an open covering with $U$ quasi-compact, $V$ affine, and $U \cap V$ quasi-compact. If approximation by perfect complexes holds on $U$, then approximation holds on $X$.
Proof. Let $T \subset X$ be a closed subset with $X \setminus T$ retro-compact in $X$. Let $r_ U$ be the integer of Definition 36.14.2 adapted to the pair $(U, T \cap U)$. Set $T' = T \setminus U$. Note that $T' \subset V$ and that $V \setminus T' = (X \setminus T) \cap U \cap V$ is quasi-compact by our assumption on $T$. Let $r'$ be the number of affines needed to cover $V \setminus T'$. 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 36.14.4 guarantees the existence of an approximation $P \to E|_ V$ of $(T', E|_ V, m)$ on $V$. Applying Lemma 36.14.3 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 $H^ t(P) \to H^ t(E|_ U)$. 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 $Q \otimes ^\mathbf {L} I$ 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 $T$ 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, Lemma 20.47.4), $H^ i(E')|_ U = 0$ 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, Lemma 20.49.7) supported on $T$. An easy diagram chase shows that $R'' \to E$ is the desired approximation. $\square$
Theorem 36.14.6. Let $X$ be a quasi-compact and quasi-separated scheme. Then approximation by perfect complexes holds on $X$.
Proof. This follows from the induction principle of Cohomology of Schemes, Lemma 30.4.1 and Lemmas 36.14.5 and 36.14.4. $\square$
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