The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

7.17 Quasi-compact objects and colimits

To be able to use the same language as in the case of topological spaces we introduce the following terminology.

Definition 7.17.1. Let $\mathcal{C}$ be a site. An object $U$ of $\mathcal{C}$ is quasi-compact if given a covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ in $\mathcal{C}$ there exists another covering $\mathcal{V} = \{ V_ j \to U\} _{j \in J}$ and a morphism $\mathcal{V} \to \mathcal{U}$ of families of maps with fixed target given by $\alpha : J \to I$ and $V_ j \to U_{\alpha (j)}$ (see Definition 7.8.1) such that the image of $\alpha $ is a finite subset of $I$.

Of course the usual notion is sufficient to conclude that $U$ is quasi-compact.

Lemma 7.17.2. Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. Consider the following conditions

  1. $U$ is quasi-compact,

  2. for every covering $\{ U_ i \to U\} _{i \in I}$ in $\mathcal{C}$ there exists a finite covering $\{ V_ j \to U\} _{j = 1, \ldots , m}$ of $\mathcal{C}$ refining $\mathcal{U}$, and

  3. for every covering $\{ U_ i \to U\} _{i \in I}$ in $\mathcal{C}$ there exists a finite subset $I' \subset I$ such that $\{ U_ i \to U\} _{i \in I'}$ is a covering in $\mathcal{C}$.

Then we always have (3) $\Rightarrow $ (2) $\Rightarrow $ (1) but the reverse implications do not hold in general.

Proof. The implications are immediate from the definitions. Let $X = [0, 1] \subset \mathbf{R}$ as a topological space (with the usual $\epsilon $-$\delta $ topology). Let $\mathcal{C}$ be the category of open subspaces of $X$ with inclusions as morphisms and usual open coverings (compare with Example 7.6.4). However, then we change the notion of covering in $\mathcal{C}$ to exclude all finite coverings, except for the coverings of the form $\{ U \to U\} $. It is easy to see that this will be a site as in Definition 7.6.2. In this site the object $X = U = [0, 1]$ is quasi-compact in the sense of Definition 7.17.1 but $U$ does not satisfy (2). We leave it to the reader to make an example where (2) holds but not (3). $\square$

Here is the topos theoretic meaning of a quasi-compact object.

Lemma 7.17.3. Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. The following are equivalent

  1. $U$ is quasi-compact, and

  2. for every surjection of sheaves $\coprod _{i \in I} \mathcal{F}_ i \to h_ U^\# $ there is a finite subset $J \subset I$ such that $\coprod _{i \in J} \mathcal{F}_ i \to h_ U^\# $ is surjective.

Proof. Assume (1) and let $\coprod _{i \in I} \mathcal{F}_ i \to h_ U^\# $ be a surjection. Then $\text{id}_ U$ is a section of $h_ U^\# $ over $U$. Hence there exists a covering $\{ U_ a \to U\} _{a \in A}$ and for each $a \in A$ a section $s_ a$ of $\coprod _{i \in I} \mathcal{F}_ i$ over $U_ a$ mapping to $\text{id}_ U$. By the construction of coproducts as sheafification of coproducts of presheaves (Lemma 7.10.13), for each $a$ there exists a covering $\{ U_{ab} \to U_ a\} _{b \in B_ a}$ and for all $b \in B_ a$ an $\iota (b) \in I$ and a section $s_{b}$ of $\mathcal{F}_{\iota (b)}$ over $U_{ab}$ mapping to $\text{id}_ U|_{U_{ab}}$. Thus after replacing the covering $\{ U_ a \to U\} _{a \in A}$ by $\{ U_{ab} \to U\} _{a \in A, b \in B_ a}$ we may assume we have a map $\iota : A \to I$ and for each $a \in A$ a section $s_ a$ of $\mathcal{F}_{\iota (a)}$ over $U_ a$ mapping to $\text{id}_ U$. Since $U$ is quasi-compact, there is a covering $\{ V_ c \to U\} _{c \in C}$, a map $\alpha : C \to A$ with finite image, and $V_ c \to U_{\alpha (c)}$ over $U$. Then we see that $J = \mathop{\mathrm{Im}}(\iota \circ \alpha ) \subset I$ works because $\coprod _{c \in C} h_{V_ c}^\# \to h_ U^\# $ is surjective (Lemma 7.12.4) and factors through $\coprod _{i \in J} \mathcal{F}_ i \to h_ U^\# $. (Here we use that the composition $h_{V_ c}^\# \to h_{U_{\alpha (c)}} \xrightarrow {s_{\alpha (c)}} \mathcal{F}_{\iota (\alpha (c))} \to h_ U^\# $ is the map $h_{V_ c}^\# \to h_ U^\# $ coming from the morphism $V_ c \to U$ because $s_{\alpha (c)}$ maps to $\text{id}_ U|_{U_{\alpha (c)}}$.)

Assume (2). Let $\{ U_ i \to U\} _{i \in I}$ be a covering. By Lemma 7.12.4 we see that $\coprod _{i \in I} h_{U_ i}^\# \to h_ U^\# $ is surjective. Thus we find a finite subset $J \subset I$ such that $\coprod _{j \in J} h_{U_ j}^\# \to h_ U^\# $ is surjective. Then arguing as above we find a covering $\{ V_ c \to U\} _{c \in C}$ of $U$ in $\mathcal{C}$ and a map $\iota : C \to J$ such that $\text{id}_ U$ lifts to a section of $s_ c$ of $h_{U_{\iota (c)}}^\# $ over $V_ c$. Refining the covering even further we may assume $s_ c \in h_{U_{\iota (c)}}(V_ c)$ mapping to $\text{id}_ U$. Then $s_ c : V_ c \to U_{\iota (c)}$ is a morphism over $U$ and we conclude. $\square$

The lemma above motivates the following definition.

Definition 7.17.4. An object $\mathcal{F}$ of a topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is quasi-compact if for any surjective map $\coprod _{i \in I} \mathcal{F}_ i \to \mathcal{F}$ of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ there exists a finite subset $J \subset I$ such that $\coprod _{i \in J} \mathcal{F}_ i \to \mathcal{F}$ is surjective. A topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is said to be quasi-compact if its final object $*$ is a quasi-compact object.

By Lemma 7.17.3 if the site $\mathcal{C}$ has a final object $X$, then $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is quasi-compact if and only if $X$ is quasi-compact. The following lemma is the analogue of Sheaves, Lemma 6.29.1 for sites.

Lemma 7.17.5. Let $\mathcal{C}$ be a site. Let $\mathcal{I} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, $i \mapsto \mathcal{F}_ i$ be a filtered diagram of sheaves of sets. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Consider the canonical map

\[ \Psi : \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i(U) \longrightarrow \left(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i\right)(U) \]

With the terminology introduced above:

  1. If all the transition maps are injective then $\Psi $ is injective for any $U$.

  2. If $U$ is quasi-compact, then $\Psi $ is injective.

  3. If $U$ is quasi-compact and all the transition maps are injective then $\Psi $ is an isomorphism.

  4. If $U$ has a cofinal system of coverings $\{ U_ j \to U\} _{j \in J}$ with $J$ finite and $U_ j \times _ U U_{j'}$ quasi-compact for all $j, j' \in J$, then $\Psi $ is bijective.

Proof. Assume all the transition maps are injective. In this case the presheaf $\mathcal{F}' : V \mapsto \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i(V)$ is separated (see Definition 7.10.9). By Lemma 7.10.13 we have $(\mathcal{F}')^\# = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$. By Theorem 7.10.10 we see that $\mathcal{F}' \to (\mathcal{F}')^\# $ is injective. This proves (1).

Assume $U$ is quasi-compact. Suppose that $s \in \mathcal{F}_ i(U)$ and $s' \in \mathcal{F}_{i'}(U)$ give rise to elements on the left hand side which have the same image under $\Psi $. This means we can choose a covering $\{ U_ a \to U\} _{a \in A}$ and for each $a \in A$ an index $i_ a \in I$, $i_ a \geq i$, $i_ a \geq i'$ such that $\varphi _{ii_ a}(s) = \varphi _{i'i_ a}(s')$. Because $U$ is quasi-compact we can choose a covering $\{ V_ b \to U\} _{b \in B}$, a map $\alpha : B \to A$ with finite image, and morphisms $V_ b \to U_{\alpha (b)}$ over $U$. Pick $i''\in I$ to be $\geq $ than all of the $i_{\alpha (b)}$ which is possible because the image of $\alpha $ is finite. We conclude that $\varphi _{ii''}(s)$ and $\varphi _{i'i''}(s)$ agree on $V_ b$ for all $b \in B$ and hence that $\varphi _{ii''}(s) = \varphi _{i'i''}(s)$. This proves (2).

Assume $U$ is quasi-compact and all transition maps injective. Let $s$ be an element of the target of $\Psi $. There exists a covering $\{ U_ a \to U\} _{a \in A}$ and for each $a \in A$ an index $i_ a \in I$ and a section $s_ a \in \mathcal{F}_{i_ a}(U_ a)$ such that $s|_{U_ a}$ comes from $s_ a$ for all $a \in A$. Because $U$ is quasi-compact we can choose a covering $\{ V_ b \to U\} _{b \in B}$, a map $\alpha : B \to A$ with finite image, and morphisms $V_ b \to U_{\alpha (b)}$ over $U$. Pick $i \in I$ to be $\geq $ than all of the $i_{\alpha (b)}$ which is possible because the image of $\alpha $ is finite. By (1) the sections $s_ b = \varphi _{i_{\alpha (b)} i}(s_{\alpha (b)})|_{V_ b}$ agree over $V_ b \times _ U V_{b'}$. Hence they glue to a section $s' \in \mathcal{F}_ i(U)$ which maps to $s$ under $\Psi $. This proves (3).

Assume the hypothesis of (4). Let $s$ be an element of the target of $\Psi $. By assumption there exists a finite covering $\{ U_ j \to U\} _{j = 1, \ldots , m} U_ j$, with $U_ j \times _ U U_{j'}$ quasi-compact for all $j, j' \in J$ and for each $j$ an index $i_ j \in I$ and $s_ j \in \mathcal{F}_{i_ j}(U_ j)$ such that $s|_{U_ j}$ is the image of $s_ j$ for all $j$. Since $U_ j \times _ U U_{j'}$ is quasi-compact we can apply (2) and we see that there exists an $i_{jj'} \in I$, $i_{jj'} \geq i_ j$, $i_{jj'} \geq i_{j'}$ such that $\varphi _{i_ ji_{jj'}}(s_ j)$ and $\varphi _{i_{j'}i_{jj'}}(s_{j'})$ agree over $U_ j \times _ U U_{j'}$. Choose an index $i \in I$ wich is bigger or equal than all the $i_{jj'}$. Then we see that the sections $\varphi _{i_ ji}(s_ j)$ of $\mathcal{F}_ i$ glue to a section of $\mathcal{F}_ i$ over $U$. This section is mapped to the element $s$ as desired. $\square$


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