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*}

18.29 Towards constructible modules

Recall that a quasi-compact object of a site is roughly an object such that every covering of it can be refined by a finite covering (the actual definition is slightly more involved, see Sites, Section 7.17). It turns out that if every object of a site has a covering by quasi-compact objects, then the modules $j_!\mathcal{O}_ U$ with $U$ quasi-compact form a particularly nice set of generators for the category of all modules.

Lemma 18.29.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\{ U_ i \to U\} $ be a covering of $\mathcal{C}$. Then the sequence

\[ \bigoplus j_{U_ i \times _ U U_ j!}\mathcal{O}_{U_ i \times _ U U_ j} \to \bigoplus j_{U_ i!}\mathcal{O}_{U_ i} \to j_!\mathcal{O}_ U \to 0 \]

is exact.

Proof. This holds because for any $\mathcal{O}$-module $\mathcal{F}$ the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ turns our sequence into the exact sequence $0 \to \mathcal{F}(U) \to \prod \mathcal{F}(U_ i) \to \prod \mathcal{F}(U_ i \times _ U U_ j)$. Then the lemma follows from Homology, Lemma 12.5.8. $\square$

Lemma 18.29.2. Let $\mathcal{C}$ be a site. Let $W$ be a quasi-compact object of $\mathcal{C}$.

  1. The functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}(W)$ commutes with coproducts.

  2. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. The functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}(W)$ commutes with direct sums.

Proof. Proof of (1). Taking sections over $W$ commutes with filtered colimits with injective transition maps by Sites, Lemma 7.17.5. If $\mathcal{F}_ i$ is a family of sheaves of sets indexed by a set $I$. Then $\coprod \mathcal{F}_ i$ is the filtered colimit over the partially ordered set of finite subsets $E \subset I$ of the coproducts $\mathcal{F}_ E = \coprod _{i \in E} \mathcal{F}_ i$. Since the transition maps are injective we conclude.

Proof of (2). Let $\mathcal{F}_ i$ be a family of sheaves of $\mathcal{O}$-modules indexed by a set $I$. Then $\bigoplus \mathcal{F}_ i$ is the filtered colimit over the partially ordered set of finite subsets $E \subset I$ of the direct sums $\mathcal{F}_ E = \bigoplus _{i \in E} \mathcal{F}_ i$. A filtered colimit of abelian sheaves can be computed in the category of sheaves of sets. Moreover, for $E \subset E'$ the transition map $\mathcal{F}_ E \to \mathcal{F}_{E'}$ is injective (as sheafification is exact and the injectivity is clear on underlying presheaves). Hence it suffices to show the result for a finite index set by Sites, Lemma 7.17.5. The finite case is dealt with in Lemma 18.3.2 (it holds over any object of $\mathcal{C}$). $\square$

Lemma 18.29.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be a quasi-compact object of $\mathcal{C}$. Then the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{O}_ U, -)$ commutes with direct sums.

Proof. This is true because $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{O}_ U, \mathcal{F}) = \mathcal{F}(U)$ and because the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ commutes with direct sums by Lemma 18.29.2. $\square$

In order to state the sharpest possible results in the following we introduce some notation.

Situation 18.29.4. Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \text{Ob}(\mathcal{C})$ be a set of objects. We consider the following conditions

  1. Every object of $\mathcal{C}$ has a covering by elements of $\mathcal{B}$.

  2. Every $U \in \mathcal{B}$ is quasi-compact in the sense that every covering of $U$ can be refined by a finite covering with objects from $\mathcal{B}$.

  3. For a finite covering $\{ U_ i \to U\} $ with $U_ i, U \in \mathcal{B}$ the fibre products $U_ i \times _ U U_ j$ are quasi-compact.

Lemma 18.29.5. In Situation 18.29.4 assume (1) holds.

  1. Every sheaf of sets is the target of a surjective map whose source is a coproduct $\coprod h_{U_ i}^\# $ with $U_ i$ in $\mathcal{B}$.

  2. If $\mathcal{O}$ is a sheaf of rings, then every $\mathcal{O}$-module is a quotient of a direct sum $\bigoplus \nolimits j_{U_ i!}\mathcal{O}_{U_ i}$ with $U_ i$ in $\mathcal{B}$.

Proof. Follows immediately from Lemmas 18.28.7 and 18.29.1. $\square$

Lemma 18.29.6. In Situation 18.29.4 assume (1) and (2) hold.

  1. Every sheaf of sets is a filtered colimit of sheaves of the form

    18.29.6.1
    \begin{equation} \label{sites-modules-equation-towards-constructible-sets} \text{Coequalizer}\left( \xymatrix{ \coprod \nolimits _{j = 1, \ldots , m} h_{V_ j}^\# \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{i = 1, \ldots , n} h_{U_ i}^\# } \right) \end{equation}

    with $U_ i$ and $V_ j$ in $\mathcal{B}$.

  2. If $\mathcal{O}$ is a sheaf of rings, then every $\mathcal{O}$-module is a filtered colimit of sheaves of the form

    18.29.6.2
    \begin{equation} \label{sites-modules-equation-towards-constructible} \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{j = 1, \ldots , m} j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i} \right) \end{equation}

    with $U_ i$ and $V_ j$ in $\mathcal{B}$.

Proof. Proof of (1). By Lemma 18.29.5 every sheaf of sets $\mathcal{F}$ is the target of a surjection whose source is a coprod $\mathcal{F}_0$ of sheaves the form $h_{U}^\# $ with $U \in \mathcal{B}$. Applying this to $\mathcal{F}_0 \times _\mathcal {F} \mathcal{F}_0$ we find that $\mathcal{F}$ is a coequalizer of a pair of maps

\[ \xymatrix{ \coprod \nolimits _{j \in J} h_{V_ j}^\# \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{i \in I} h_{U_ i}^\# } \]

for some index sets $I$, $J$ and $V_ j$ and $U_ i$ in $\mathcal{B}$. For every finite subset $J' \subset J$ there is a finite subset $I' \subset I$ such that the coproduct over $j \in J'$ maps into the coprod over $i \in I'$ via both maps, see Lemma 18.29.3. Thus our sheaf is the colimit of the cokernels of these maps between finite coproducts.

Proof of (2). By Lemma 18.29.5 every module is a quotient of a direct sum of modules of the form $j_{U!}\mathcal{O}_ U$ with $U \in \mathcal{B}$. Thus every module is a cokernel

\[ \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{j \in J} j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus \nolimits _{i \in I} j_{U_ i!}\mathcal{O}_{U_ i} \right) \]

for some index sets $I$, $J$ and $V_ j$ and $U_ i$ in $\mathcal{B}$. For every finite subset $J' \subset J$ there is a finite subset $I' \subset I$ such that the direct sum over $j \in J'$ maps into the direct sum over $i \in I'$, see Lemma 18.29.3. Thus our module is the colimit of the cokernels of these maps between finite direct sums. $\square$

Proof. Let $\mathcal{F} = \mathop{\mathrm{Coker}}(\bigoplus j_{V_ j!}\mathcal{O}_{V_ j} \to \bigoplus j_{U_ i!}\mathcal{O}_{U_ i})$ as in (18.29.6.2). It suffices to show that the cokernel of a map $\varphi : j_{W!}\mathcal{O}_ W \to \mathcal{F}$ with $W \in \mathcal{B}$ is another module of the same type. The map $\varphi $ corresponds to $s \in \mathcal{F}(W)$. By (2) we can find a finite covering $\{ W_ k \to W\} $ with $W_ k \in \mathcal{B}$ such that $s|_{W_ k}$ comes from a section $\sum s_{ki}$ of $\bigoplus j_{U_ i!}\mathcal{O}_{U_ i})$. This determines maps $j_{W_ k!}\mathcal{O}_{W_ k} \to \bigoplus j_{U_ i!}\mathcal{O}_{U_ i}$. Since $\bigoplus j_{W_ k!}\mathcal{O}_{W_ k} \to j_{W!}\mathcal{O}_ W$ is surjective (Lemma 18.29.1) we see that $\mathop{\mathrm{Coker}}(\varphi )$ is equal to

\[ \mathop{\mathrm{Coker}}\left( \bigoplus j_{W_ k!}\mathcal{O}_{W_ k} \oplus \bigoplus j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus j_{U_ i!}\mathcal{O}_{U_ i} \right) \]

as desired. $\square$

Lemma 18.29.8. In Situation 18.29.4 assume (1), (2), and (3) hold. Let $\mathcal{O}$ be a sheaf of rings. Then given a map

\[ \bigoplus \nolimits _{j = 1, \ldots , m} j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i} \]

with $U_ i$ and $V_ j$ in $\mathcal{B}$, and finite coverings $\{ U_{ik} \to U_ i\} $ by $U_{ik} \in \mathcal{B}$, there exist a finite set of $W_ l \in \mathcal{B}$ and a commutative diagram

\[ \xymatrix{ \bigoplus j_{W_ l!}\mathcal{O}_{W_ l} \ar[d] \ar[r] & \bigoplus j_{U_ i!}\mathcal{O}_{U_{ik}} \ar[d] \\ \bigoplus j_{V_ j!}\mathcal{O}_{V_ j} \ar[r] & \bigoplus j_{U_ i!}\mathcal{O}_{U_ i} } \]

inducing an isomorphism on cokernels of the horizontal maps.

Proof. Since $\bigoplus j_{U_{ik}!}\mathcal{O}_{U_{ik}} \to \bigoplus j_{U_ i!}\mathcal{O}_{U_ i}$ is surjective (Lemma 18.29.1), we can find finite coverings $\{ V_{jm} \to V_ j\} $ with $V_{jm} \in \mathcal{B}$ such that we can find a commutative diagram

\[ \xymatrix{ \bigoplus j_{V_{jm}!}\mathcal{O}_{V_{jm}} \ar[d] \ar[r] & \bigoplus j_{U_ i!}\mathcal{O}_{U_{ik}} \ar[d] \\ \bigoplus j_{V_ j!}\mathcal{O}_{V_ j} \ar[r] & \bigoplus j_{U_ i!}\mathcal{O}_{U_ i} } \]

Adding

\[ \bigoplus j_{U_{ik} \times _{U_ i} U_{ik'}!}\mathcal{O}_{U_{ik} \times _{U_ i} U_{ik'}} \]

to the upper left corner finishes the proof by Lemma 18.29.1. $\square$

Proof. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of $\mathcal{O}$-modules with $\mathcal{F}_1$ and $\mathcal{F}_3$ as in (18.29.6.2). Choose presentations

\[ \bigoplus A_{V_ j} \to \bigoplus A_{U_ i} \to \mathcal{F}_1 \to 0 \quad \text{and}\quad \bigoplus A_{T_ j} \to \bigoplus A_{W_ i} \to \mathcal{F}_3 \to 0 \]

In this proof the direct sums are always finite, and we write $A_ U = j_{U!}\mathcal{O}_ U$ for $U \in \mathcal{B}$. By Lemma 18.29.8 we may replace $W_ i$ by finite coverings $\{ W_{ik} \to W_ i\} $ with $W_{ik} \in \mathcal{B}$. Thus we may assume the map $\bigoplus A_{W_ i} \to \mathcal{F}_3$ lifts to a map into $\mathcal{F}_2$. Consider the kernel

\[ \mathcal{K}_2 = \mathop{\mathrm{Ker}}(\bigoplus A_{U_ i} \oplus \bigoplus A_{W_ i} \longrightarrow \mathcal{F}_2) \]

By the snake lemma this kernel surjections onto $\mathcal{K}_3 = \mathop{\mathrm{Ker}}(\bigoplus A_{W_ i} \to \mathcal{F}_3)$. Thus after replacing each $T_ j$ by a finite covering with elements of $\mathcal{B}$ (permissible by Lemma 18.29.1) we may assume there is a map $\bigoplus A_{T_ j} \to \mathcal{K}_2$ lifting the given map $\bigoplus A_{T_ j} \to \mathcal{K}_3$. Then $\bigoplus A_{V_ j} \oplus \bigoplus A_{T_ j} \to \mathcal{K}_2$ is surjective which finishes the proof. $\square$

Lemma 18.29.10. In Situation 18.29.4 assume (1), (2), and (3) hold. Let $\mathcal{O}$ be a sheaf of rings. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ be the full subcategory of modules isomorphic to a cokernel as in (18.29.6.2). If the kernel of every map of $\mathcal{O}$-modules of the form

\[ \bigoplus \nolimits _{j = 1, \ldots , m} j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i} \]

with $U_ i$ and $V_ j$ in $\mathcal{B}$, is in $\mathcal{A}$, then $\mathcal{A}$ is weak Serre subcategory of $\textit{Mod}(\mathcal{O})$.

Proof. We will use the criterion of Homology, Lemma 12.9.3. By the results of Lemmas 18.29.7 and 18.29.9 it suffices to see that the kernel of a map $\mathcal{F} \to \mathcal{G}$ between objects of $\mathcal{A}$ is in $\mathcal{A}$. To prove this choose presentations

\[ \bigoplus A_{V_ j} \to \bigoplus A_{U_ i} \to \mathcal{F} \to 0 \quad \text{and}\quad \bigoplus A_{T_ j} \to \bigoplus A_{W_ i} \to \mathcal{G} \to 0 \]

In this proof the direct sums are always finite, and we write $A_ U = j_{U!}\mathcal{O}_ U$ for $U \in \mathcal{B}$. Using Lemmas 18.29.1 and 18.29.8 and arguing as in the proof of Lemma 18.29.9 we may assume that the map $\mathcal{F} \to \mathcal{G}$ lifts to a map of presentations

\[ \xymatrix{ \bigoplus A_{V_ j} \ar[r] \ar[d] & \bigoplus A_{U_ i} \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ \bigoplus A_{T_ j} \ar[r] & \bigoplus A_{W_ i} \ar[r] & \mathcal{G} \ar[r] & 0 } \]

Then we see that

\[ \mathop{\mathrm{Ker}}(\mathcal{F} \to \mathcal{G}) = \mathop{\mathrm{Coker}}\left(\bigoplus A_{V_ j} \to \mathop{\mathrm{Ker}}\left( \bigoplus A_{T_ j} \oplus \bigoplus A_{U_ i} \to \bigoplus A_{W_ i}\right)\right) \]

and the lemma follows from the assumption and Lemma 18.29.7. $\square$


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