The Stacks project

Proof. We will use the induction principle of Lemma 75.9.6 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 75.4.2 (and the compatibilities of Remark 75.6.3) for the representable algebraic spaces $V$ and $W \times _ X V$. We will also use the fact that perfectness on the Zariski site and étale site agree, see Lemma 75.13.5.

By Derived Categories of Schemes, Lemma 36.13.10 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 36.13.7 we can replace $Q$ by $Q \otimes ^\mathbf {L} I$ (still supported on $f^{-1}T$) and assume that the map

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 $a|_ W$ 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. $\square$

Proof. Special case of Lemma 75.15.1. $\square$

Proof. We will prove this using the induction principle of Lemma 75.9.3.

If $X$ is affine, then $\mathcal{O}_ X$ is a perfect generator. This follows from Lemma 75.4.2 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 75.15.3 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 75.9.1). 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 75.14.3 and 75.10.7). 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 75.10.7 we have $K' = f_! K$ and hence

Thus by Derived Categories of Schemes, Lemma 36.15.2 (using also Lemma 75.4.2) 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

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. $\square$

Proof. We will prove this using the induction principle of Lemma 75.9.3. 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 36.15.4.

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 75.15.1 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 75.14.3 and 75.10.7). 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 75.10.7 we have $K' = R(V \to X)_! K$ and hence

Thus by Derived Categories of Schemes, Lemma 36.15.2 we have $E|_ V = Rj_*E|_ W$ where $j : W \to V$ is the inclusion. Picture

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

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 75.6.3. This implies that $E = R(U \to X)_*E|_ U$ (small detail omitted). If this is the case then

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. $\square$