The Stacks project

72.15 Generating derived categories

This section is the analogue of Derived Categories of Schemes, Section 36.14. However, we first prove the following lemma which is the analogue of Derived Categories of Schemes, Lemma 36.12.9.

Lemma 72.15.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $W \subset X$ be a quasi-compact open. Let $T \subset |X|$ be a closed subset such that $X \setminus T \to X$ is a quasi-compact morphism. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $\alpha : P \to E|_ W$ be a map where $P$ is a perfect object of $D(\mathcal{O}_ W)$ supported on $T \cap W$. 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 $R|_ W$ in $D(\mathcal{O}_ W)$ compatible $\alpha $ and $\beta |_ W$.

Proof. We will use the induction principle of Lemma 72.9.6 to prove this. Thus we immediately reduce to the case where we have an elementary distinguished square $(W \subset X, f : V \to X)$ with $V$ affine and $P \to E|_ W$ as in the statement of the lemma. In the rest of the proof we will use Lemma 72.4.2 (and the compatibilities of Remark 72.6.3) for the representable algebraic spaces $V$ and $W \times _ X V$. We will also use the fact that perfectness on the Zariski site and ├ętale site agree, see Lemma 72.13.5.

By Derived Categories of Schemes, Lemma 36.12.8 we can choose a perfect object $Q$ in $D(\mathcal{O}_ V)$ supported on $f^{-1}T$ and an isomorphism $Q|_{W \times _ X V} \to (P \oplus P[1])|_{W \times _ X V}$. By Derived Categories of Schemes, Lemma 36.12.5 we can replace $Q$ by $Q \otimes ^\mathbf {L} I$ (still supported on $f^{-1}T$) and assume that the map

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

lifts to $Q \to E|_ V$. By Lemma 72.10.8 we find an morphism $a : R \to E$ of $D(\mathcal{O}_ X)$ such that $a|_ W$ is isomorphic to $P \oplus P[1] \to E|_ W$ and $a|_ V$ isomorphic to $Q \to E|_ V$. Thus $R$ is perfect and supported on $T$ as desired. $\square$

Remark 72.15.2. The proof of Lemma 72.15.1 shows that

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

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

Lemma 72.15.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $W$ be a quasi-compact open subspace of $X$. Let $P$ be a perfect object of $D(\mathcal{O}_ W)$. Then $P$ is a direct summand of the restriction of a perfect object of $D(\mathcal{O}_ X)$.

Proof. Special case of Lemma 72.15.1. $\square$

Theorem 72.15.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The category $D_\mathit{QCoh}(\mathcal{O}_ X)$ can be generated by a single perfect object. More precisely, there exists a perfect object $P$ of $D(\mathcal{O}_ X)$ such that for $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$ the following are equivalent

  1. $E = 0$, and

  2. $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[n], E) = 0$ for all $n \in \mathbf{Z}$.

Proof. We will prove this using the induction principle of Lemma 72.9.3.

If $X$ is affine, then $\mathcal{O}_ X$ is a perfect generator. This follows from Lemma 72.4.2 and Derived Categories of Schemes, Lemma 36.3.5.

Assume that $(U \subset X, f : V \to X)$ is an elementary distinguished square with $U$ quasi-compact such that the theorem holds for $U$ and $V$ is an affine scheme. Let $P$ be a perfect object of $D(\mathcal{O}_ U)$ which is a generator for $D_\mathit{QCoh}(\mathcal{O}_ U)$. Using Lemma 72.15.3 we may choose a perfect object $Q$ of $D(\mathcal{O}_ X)$ whose restriction to $U$ is a direct sum one of whose summands is $P$. Say $V = \mathop{\mathrm{Spec}}(A)$. Let $Z \subset V$ be the reduced closed subscheme which is the inverse image of $X \setminus U$ and maps isomorphically to it (see Definition 72.9.1). This is a retrocompact closed subset of $V$. Choose $f_1, \ldots , f_ r \in A$ such that $Z = V(f_1, \ldots , f_ r)$. Let $K \in D(\mathcal{O}_ V)$ be the perfect object corresponding to the Koszul complex on $f_1, \ldots , f_ r$ over $A$. Note that since $K$ is supported on $Z$, the pushforward $K' = Rf_*K$ is a perfect object of $D(\mathcal{O}_ X)$ whose restriction to $V$ is $K$ (see Lemmas 72.14.3 and 72.10.7). We claim that $Q \oplus K'$ is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ such that there are no nontrivial maps from any shift of $Q \oplus K'$ into $E$. By Lemma 72.10.7 we have $K' = f_! K$ and hence

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K'[n], E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ V)}(K[n], E|_ V) \]

Thus by Derived Categories of Schemes, Lemma 36.14.2 (using also Lemma 72.4.2) the vanishing of these groups implies that $E|_ V$ is isomorphic to $R(U \times _ X V \to V)_*E|_{U \times _ X V}$. This implies that $E = R(U \to X)_*E|_ U$ (small detail omitted). If this is the case then

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(Q[n], E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(Q|_ U[n], E|_ U) \]

which contains $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(P[n], E|_ U)$ as a direct summand. Thus by our choice of $P$ the vanishing of these groups implies that $E|_ U$ is zero. Whence $E$ is zero. $\square$

Remark 72.15.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ be a generator (see Theorem 72.15.4). Then the following are equivalent

  1. for $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $Rf_*K = 0$ if and only if $K = 0$,

  2. $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ reflects isomorphisms, and

  3. $Lf^*E$ is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$.

The equivalence between (1) and (2) is a formal consequence of the fact that $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ is an exact functor of triangulated categories. Similarly, the equivalence between (1) and (3) follows formally from the fact that $Lf^*$ is the left adjoint to $Rf_*$. These conditions hold if $f$ is affine (Lemma 72.6.4) or if $f$ is an open immersion, or if $f$ is a composition of such.

The following result is an strengthening of Theorem 72.15.4 proved using exactly the same methods. Let $T \subset |X|$ be a closed subset where $X$ is an algebraic space. Let's denote $D_ T(\mathcal{O}_ X)$ the strictly full, saturated, triangulated subcategory consisting of complexes whose cohomology sheaves are supported on $T$.

Lemma 72.15.6. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \subset |X|$ be a closed subset such that $|X| \setminus T$ is quasi-compact. With notation as above, the category $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ is generated by a single perfect object.

Proof. We will prove this using the induction principle of Lemma 72.9.3. The property is true for representable quasi-compact and quasi-separated objects of the site $X_{spaces, {\acute{e}tale}}$ by Derived Categories of Schemes, Lemma 36.14.4.

Assume that $(U \subset X, f : V \to X)$ is an elementary distinguished square such that the lemma holds for $U$ and $V$ is affine. To finish the proof we have to show that the result holds for $X$. Let $P$ be a perfect object of $D(\mathcal{O}_ U)$ supported on $T \cap U$ which is a generator for $D_{\mathit{QCoh}, T \cap U}(\mathcal{O}_ U)$. Using Lemma 72.15.1 we may choose a perfect object $Q$ of $D(\mathcal{O}_ X)$ supported on $T$ whose restriction to $U$ is a direct sum one of whose summands is $P$. Write $V = \mathop{\mathrm{Spec}}(B)$. Let $Z = X \setminus U$. Then $f^{-1}Z$ is a closed subset of $V$ such that $V \setminus f^{-1}Z$ is quasi-compact. As $X$ is quasi-separated, it follows that $f^{-1}Z \cap f^{-1}T = f^{-1}(Z \cap T)$ is a closed subset of $V$ such that $W = V \setminus f^{-1}(Z \cap T)$ is quasi-compact. Thus we can choose $g_1, \ldots , g_ s \in B$ such that $f^{-1}(Z \cap T) = V(g_1, \ldots , g_ r)$. Let $K \in D(\mathcal{O}_ V)$ be the perfect object corresponding to the Koszul complex on $g_1, \ldots , g_ s$ over $B$. Note that since $K$ is supported on $f^{-1}(Z \cap T) \subset V$ closed, the pushforward $K' = R(V \to X)_*K$ is a perfect object of $D(\mathcal{O}_ X)$ whose restriction to $V$ is $K$ (see Lemmas 72.14.3 and 72.10.7). We claim that $Q \oplus K'$ is a generator for $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$.

Let $E$ be an object of $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ such that there are no nontrivial maps from any shift of $Q \oplus K'$ into $E$. By Lemma 72.10.7 we have $K' = R(V \to X)_! K$ and hence

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K'[n], E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ V)}(K[n], E|_ V) \]

Thus by Derived Categories of Schemes, Lemma 36.14.2 we have $E|_ V = Rj_*E|_ W$ where $j : W \to V$ is the inclusion. Picture

\[ \xymatrix{ W \ar[r]_ j & V & Z \cap T \ar[l] \ar[d] \\ V \setminus f^{-1}Z \ar[u]^{j'} \ar[ru]_{j''} & & Z \ar[lu] } \]

Since $E$ is supported on $T$ we see that $E|_ W$ is supported on $f^{-1}T \cap W = f^{-1}T \cap (V \setminus f^{-1}Z)$ which is closed in $W$. We conclude that

\[ E|_ V = Rj_*(E|_ W) = Rj_*(Rj'_*(E|_{U \cap V})) = Rj''_*(E|_{U \cap V}) \]

Here the second equality is part (1) of Cohomology, Lemma 20.33.6 which applies because $V$ is a scheme and $E$ has quasi-coherent cohomology sheaves hence pushforward along the quasi-compact open immersion $j'$ agrees with pushforward on the underlying schemes, see Remark 72.6.3. This implies that $E = R(U \to X)_*E|_ U$ (small detail omitted). If this is the case then

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(Q[n], E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(Q|_ U[n], E|_ U) \]

which contains $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(P[n], E|_ U)$ as a direct summand. Thus by our choice of $P$ the vanishing of these groups implies that $E|_ U$ is zero. Whence $E$ is zero. $\square$


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