The Stacks project

Proof. The final object is the singleton space. Given morphisms $X \to Z$ and $Y \to Z$ of $\textit{LC}$ the fibre product $X \times _ Z Y$ is a subspace of $X \times Y$. Hence $X \times _ Z Y$ is Hausdorff as $X \times Y$ is Hausdorff by Topology, Section 5.3.

If $X$ and $Y$ are quasi-compact, then $X \times Y$ is quasi-compact by Topology, Theorem 5.14.4. Since $X \times _ Z Y$ is a closed subset of $X \times Y$ (Topology, Lemma 5.3.4) we find that $X \times _ Z Y$ is quasi-compact by Topology, Lemma 5.12.3.

Finally, returning to the general case, if $x \in X$ and $y \in Y$ we can pick quasi-compact neighbourhoods $x \in E \subset X$ and $y \in F \subset Y$ and we find that $E \times _ Z F$ is a quasi-compact neighbourhood of $(x, y)$ by the result above. Thus $X \times _ Z Y$ is an object of $\textit{LC}$ by Topology, Lemma 5.13.2. $\square$

Proof. Part (1) holds by the remark above that open coverings are qc coverings.

Proof of (2). Let $x \in X$. Choose $i_1, \ldots , i_ n \in I$ and $E_ a \subset X_{i_ a}$ quasi-compact such that $\bigcup f_{i_ a}(E_ a)$ is a neighbourhood of $x$. For every $e \in E_ a$ we can find a finite subset $J_ e \subset J_{i_ a}$ and quasi-compact $F_{e, j} \subset X_{ij}$, $j \in J_ e$ such that $\bigcup g_{ij}(F_{e, j})$ is a neighbourhood of $e$. Since $E_ a$ is quasi-compact we find a finite collection $e_1, \ldots , e_{m_ a}$ such that

Then we find that

is a neighbourhood of $x$.

Proof of (3). Let $x' \in X'$ be a point. Let $x \in X$ be its image. Choose $i_1, \ldots , i_ n \in I$ and quasi-compact subsets $E_ j \subset X_{i_ j}$ such that $\bigcup f_{i_ j}(E_ j)$ is a neighbourhood of $x$. Choose a quasi-compact neighbourhood $F \subset X'$ of $x'$ which maps into the quasi-compact neighbourhood $\bigcup f_{i_ j}(E_ j)$ of $x$. Then $F \times _ X E_ j \subset X' \times _ X X_{i_ j}$ is a quasi-compact subset and $F$ is the image of the map $\coprod F \times _ X E_ j \to F$. Hence the base change is a qc covering and the proof is finished. $\square$

Proof. Let $y \in Y$ be a point. For each $x \in X_ y$ choose a quasi-compact neighbourhood $E_ x \subset X$. Choose $x \in U_ x \subset E_ x$ open. Since $f$ is proper the fibre $X_ y$ is quasi-compact and we find $x_1, \ldots , x_ n \in X_ y$ such that $X_ y \subset U_{x_1} \cup \ldots \cup U_{x_ n}$. We claim that $f(E_{x_1}) \cup \ldots \cup f(E_{x_ n})$ is a neighbourhood of $y$. Namely, as $f$ is closed (Topology, Theorem 5.17.5) we see that $Z = f(X \setminus U_{x_1} \cup \ldots \cup U_{x_ n})$ is a closed subset of $Y$ not containing $y$. As $f$ is surjective we see that $Y \setminus Z$ is contained in $f(E_{x_1}) \cup \ldots \cup f(E_{x_ n})$ as desired. $\square$

Proof. Denote $\mathcal{G}$ the presheaf given by the formula in the lemma. Of course the pullback $f^{-1}$ in the formula denotes usual pullback of sheaves on topological spaces. It is immediate from the definitions that $\mathcal{G}$ is a sheaf for the Zar topology.

Let $Y \to X$ be a morphism in $\textit{LC}_{qc}$. Let $\mathcal{V} = \{ g_ i : Y_ i \to Y\} _{i \in I}$ be a qc covering. To prove $\mathcal{G}$ is a sheaf for the qc topology it suffices to show that $\mathcal{G}(Y) \to H^0(\mathcal{V}, \mathcal{G})$ is an isomorphism, see Sites, Section 7.10. We first point out that the map is injective as a qc covering is surjective and we can detect equality of sections at stalks (use Sheaves, Lemmas 6.11.1 and 6.21.4). Thus $\mathcal{G}$ is a separated presheaf on $\textit{LC}_{qc}$ hence it suffices to show that any element $(s_ i) \in H^0(\mathcal{V}, \mathcal{G})$ maps to an element in the image of $\mathcal{G}(Y)$ after replacing $\mathcal{V}$ by a refinement (Sites, Theorem 7.10.10).

Identifying sheaves on $Y_{i, Zar}$ and sheaves on $Y_ i$ we find that $\mathcal{G}|_{Y_{i, Zar}}$ is the pullback of $f^{-1}\mathcal{F}$ under the continuous map $g_ i : Y_ i \to Y$. Thus we can choose an open covering $Y_ i = \bigcup V_{ij}$ such that for each $j$ there is an open $W_{ij} \subset Y$ and a section $t_{ij} \in \mathcal{G}(W_{ij})$ such that $V_{ij}$ maps into $W_{ij}$ and such that $s|_{V_{ij}}$ is the pullback of $t_{ij}$. In other words, after refining the covering $\{ Y_ i \to Y\} $ we may assume there are opens $W_ i \subset Y$ such that $Y_ i \to Y$ factors through $W_ i$ and sections $t_ i$ of $\mathcal{G}$ over $W_ i$ which restrict to the given sections $s_ i$. Moreover, if $y \in Y$ is in the image of both $Y_ i \to Y$ and $Y_ j \to Y$, then the images $t_{i, y}$ and $t_{j, y}$ in the stalk $f^{-1}\mathcal{F}_ y$ agree (because $s_ i$ and $s_ j$ agree over $Y_ i \times _ Y Y_ j$). Thus for $y \in Y$ there is a well defined element $t_ y$ of $f^{-1}\mathcal{F}_ y$ agreeing with $t_{i, y}$ whenever $y$ is in the image of $Y_ i \to Y$. We will show that the element $(t_ y)$ comes from a global section of $f^{-1}\mathcal{F}$ over $Y$ which will finish the proof of the lemma.

It suffices to show that this is true locally on $Y$, see Sheaves, Section 6.17. Let $y_0 \in Y$. Pick $i_1, \ldots , i_ n \in I$ and quasi-compact subsets $E_ j \subset Y_{i_ j}$ such that $\bigcup g_{i_ j}(E_ j)$ is a neighbourhood of $y_0$. Let $V \subset Y$ be an open neighbourhood of $y_0$ contained in $\bigcup g_{i_ j}(E_ j)$ and contained in $W_{i_1} \cap \ldots \cap W_{i_ n}$. Since $t_{i_1, y_0} = \ldots = t_{i_ n, y_0}$, after shrinking $V$ we may assume the sections $t_{i_ j}|_ V$, $j = 1, \ldots , n$ of $f^{-1}\mathcal{F}$ agree. As $V \subset \bigcup g_{i_ j}(E_ j)$ we see that $(t_ y)_{y \in V}$ comes from this section.

We still have to show that $\mathcal{G}$ is equal to $\epsilon _ X^{-1}\pi _ X^{-1}\mathcal{F}$ on $\textit{LC}_{qc}$, resp. $\pi _ X^{-1}\mathcal{F}$ on $\textit{LC}_{Zar}$. In both cases the pullback is defined by taking the presheaf

and then sheafifying. Sheafifying in the Zar topology exactly produces our sheaf $\mathcal{G}$ and the fact that $\mathcal{G}$ is a qc sheaf, shows that it works as well in the qc topology. $\square$

Proof. Proof of (1). The equality $H^ n_{Zar}(X, \pi _ X^{-1}\mathcal{F}) = H^ n(X, \mathcal{F})$ is a general fact coming from the trivial observation that coverings of $X$ in $\textit{LC}_{Zar}$ are the same thing as open coverings of $X$. The reader who wishes to see a detailed proof should apply Lemma 21.7.2 to the functor $X_{Zar} \to \textit{LC}_{Zar}$.

Proof of (2). This is true because $\pi _{X, *} = \tau _ X^{-1}$ for some morphism of topoi $\tau _ X : \mathop{\mathit{Sh}}\nolimits (X_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (\textit{LC}_{Zar})$ as follows from Sites, Lemma 7.21.8 applied to the functor $X_{Zar} \to \textit{LC}_{Zar}/X$ used to define $\pi _ X$.

Proof of (3). This is true because $\tau _ X^{-1} \circ \pi _ X^{-1}$ is the identity functor by Sites, Lemma 7.21.8. Or you can deduce it from the explicit description of $\pi _ X^{-1}$ in Lemma 21.31.6.

Proof of (4). Apply (3) to an complex of abelian sheaves representing $K$.

Proof of (5). The morphism of topoi $f_{Zar}$ comes from an application of Sites, Lemma 7.25.8 and in our case comes from the continuous functor $Z/Y \mapsto Z \times _ Y X/X$ by Sites, Lemma 7.27.3. The diagram commutes simply because the corresponding continuous functors compose correctly (see Sites, Lemma 7.14.4).

Proof of (6). We have $H^ n_{Zar}(X, \pi _ Y^{-1}\mathcal{G}) = H^ n_{Zar}(X, f_{Zar}^{-1}\pi _ Y^{-1}\mathcal{G})$ for $\mathcal{G}$ in $\textit{Ab}(Y)$, see Lemma 21.7.1. This is equal to $H^ n_{Zar}(X, \pi _ X^{-1}f^{-1}\mathcal{G})$ by the commutativity of the diagram in (5). Hence we conclude by (1) in the case $L$ consists of a single sheaf in degree $0$. The general case follows by representing $L$ by a bounded below complex of abelian sheaves.

Proof of (7a). Let $\mathcal{F}$ be a sheaf on $X$. Let $g : Z \to Y$ be an object of $\textit{LC}_{Zar}/Y$. Consider the fibre product

Then we have

the second equality by Lemma 21.31.6. On the other hand

again by Lemma 21.31.6. Hence by proper base change for sheaves of sets (Cohomology, Lemma 20.18.3) we conclude the two sets are canonically isomorphic. The isomorphism is compatible with restriction mappings and defines an isomorphism $\pi _ Y^{-1}f_*\mathcal{F} = f_{Zar, *}\pi _ X^{-1}\mathcal{F}$. Thus an isomorphism of functors $\pi _ Y^{-1} \circ f_* = f_{Zar, *} \circ \pi _ X^{-1}$.

Proof of (7b). Let $K \in D^+(X)$. By Lemma 21.20.6 the $n$th cohomology sheaf of $Rf_{Zar, *}\pi _ X^{-1}K$ is the sheaf associated to the presheaf

with notation as above. Observe that

The first equality is (6) applied to $K$ and $g' : Z' \to X$. The second equality is Leray for $f' : Z' \to Z$ (Cohomology, Lemma 20.13.1). The third equality is the proper base change theorem (Cohomology, Theorem 20.18.2). The fourth equality is (6) applied to $g : Z \to Y$ and $Rf_*K$. Thus $Rf_{Zar, *}\pi _ X^{-1}K$ and $\pi _ Y^{-1}Rf_*K$ have the same cohomology sheaves. We omit the verification that the canonical base change map $\pi _ Y^{-1}Rf_*K \to Rf_{Zar, *}\pi _ X^{-1}K$ induces this isomorphism. $\square$

Proof. The morphism of topoi $f_{qc}$ is the one from Sites, Lemma 7.25.8 which in our case comes from the continuous functor $Z/Y \mapsto Z \times _ Y X/X$, see Sites, Lemma 7.27.3. The diagram on the left commutes because the corresponding continuous functors compose correctly (see Sites, Lemma 7.14.4). The diagram on the right commutes because the one on the left does and because of part (5) of Lemma 21.31.7.

Proof of the final assertion. The reader may repeat the proof of part (7a) of Lemma 21.31.7; we will instead deduce this from it. As $\epsilon _{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. The description in Lemma 21.31.6 shows that $\epsilon _{Y, *} \circ a_ Y^{-1} = \pi _ Y^{-1}$ and similarly for $X$. To show that the canonical map $a_ Y^{-1}f_*\mathcal{F} \to f_{qc, *}a_ X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that

is an isomorphism. This is part (7a) of Lemma 21.31.7. $\square$

Proof. We first show that $\mathcal{A}'_ X \subset \textit{Ab}(\textit{LC}_{Zar}/X)$ is a weak Serre subcategory by checking conditions (1), (2), (3), and (4) of Homology, Lemma 12.10.3. Parts (1), (2), (3) are immediate as $\pi _ X^{-1}$ is exact and fully faithful by Lemma 21.31.7 part (3). If $0 \to \pi _ X^{-1}\mathcal{F} \to \mathcal{G} \to \pi _ X^{-1}\mathcal{F}' \to 0$ is a short exact sequence in $\textit{Ab}(\textit{LC}_{Zar}/X)$ then $0 \to \mathcal{F} \to \pi _{X, *}\mathcal{G} \to \mathcal{F}' \to 0$ is exact by Lemma 21.31.7 part (2). Hence $\mathcal{G} = \pi _ X^{-1}\pi _{X, *}\mathcal{G}$ is in $\mathcal{A}'_ X$ which checks the final condition.

Property (1) holds by Lemma 21.31.1 and the fact that the base change of a proper map is a proper map (see Topology, Theorem 5.17.5 and Lemma 5.4.4).

Property (2) follows from the commutative diagram (5) in Lemma 21.31.7.

Property (3) is Lemma 21.31.6.

Property (4) is Lemma 21.31.7 part (7)(b).

Proof of (5). Suppose given a qc covering $\{ U_ i \to U\} $. For $u \in U$ pick $i_1, \ldots , i_ m \in I$ and quasi-compact subsets $E_ j \subset U_{i_ j}$ such that $\bigcup f_{i_ j}(E_ j)$ is a neighbourhood of $u$. Observe that $Y = \coprod _{j = 1, \ldots , m} E_ j \to U$ is proper as a continuous map between Hausdorff quasi-compact spaces (Topology, Lemma 5.17.7). Choose an open neighbourhood $u \in V$ contained in $\bigcup f_{i_ j}(E_ j)$. Then $Y \times _ U V \to V$ is a surjective proper morphism and hence a $qc$ covering by Lemma 21.31.4. Since we can do this for every $u \in U$ we see that (5) holds. $\square$

Proof. By Lemma 21.31.9 the lemmas in Section 21.30 all apply to our current setting. To translate the results observe that the category $\mathcal{A}_ X$ of Lemma 21.30.2 is the essential image of $a_ X^{-1} : \textit{Ab}(X) \to \textit{Ab}(\textit{LC}_{qc}/X)$.

Part (1) is equivalent to $(V_ n)$ for all $n$ which holds by Lemma 21.30.8.

Part (2) follows by applying $\epsilon _ Y^{-1}$ to the conclusion of Lemma 21.30.3.

Part (3) follows from Lemma 21.30.8 part (1) because $\pi _ X^{-1}K$ is in $D^+_{\mathcal{A}'_ X}(\textit{LC}_{Zar}/X)$ and $a_ X^{-1} = \epsilon _ X^{-1} \circ a_ X^{-1}$.

Part (4) follows from Lemma 21.30.8 part (2) for the same reason. $\square$

Proof. We first reduce the statement to the case where $K$ is given by a single abelian sheaf. Namely, represent $K$ by a bounded below complex $\mathcal{F}^\bullet $. By the case of a sheaf we see that $\mathcal{F}^ n = a_{X, *} a_ X^{-1} \mathcal{F}^ n$ and that the sheaves $R^ qa_{X, *}a_ X^{-1}\mathcal{F}^ n$ are zero for $q > 0$. By Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) applied to $a_ X^{-1}\mathcal{F}^\bullet $ and the functor $a_{X, *}$ we conclude. From now on assume $K = \mathcal{F}$.

By Lemma 21.31.6 we have $a_{X, *}a_ X^{-1}\mathcal{F} = \mathcal{F}$. Thus it suffices to show that $R^ qa_{X, *}a_ X^{-1}\mathcal{F} = 0$ for $q > 0$. For this we can use $a_ X = \epsilon _ X \circ \pi _ X$ and the Leray spectral sequence Lemma 21.14.7. By Lemma 21.31.10 we have $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$ and $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$. By Lemma 21.31.7 we have $R^ j\pi _{X, *}(\pi _ X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof. $\square$

Proof. This follows from Lemma 21.31.11 by Remark 21.14.4. $\square$