18.30 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 18.30.1. 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. For any $\mathcal{O}$-module $\mathcal{F}$ the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ turns our sequence into the exact sequence $0 \to \mathcal{F}(U) \to \prod \mathcal{F}(U_ i) \to \prod \mathcal{F}(U_ i \times _ U U_ j)$, see (18.19.2.1). The lemma follows from this and Homology, Lemma 12.5.8. $\square$

Lemma 18.30.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be covering of $\mathcal{C}$. If $U$ is quasi-compact, then there exist a finite subset $I' \subset I$ such that the sequence

$\bigoplus \nolimits _{i, i' \in I'} j_{U_ i \times _ U U_{i'}!}\mathcal{O}_{U_ i \times _ U U_{i'}} \to \bigoplus \nolimits _{i \in I'} j_{U_ i!}\mathcal{O}_{U_ i} \to j_!\mathcal{O}_ U \to 0$

is exact.

Proof. This lemma is immediate from Lemma 18.30.1 if $U$ satisfies condition (3) of Sites, Lemma 7.17.2. We urge the reader to skip the proof in the general case. By definition there exists a 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 $\text{id} : U \to U$, $\alpha : J \to I$, and $f_ j : V_ j \to U_{\alpha (j)}$ (see Sites, Definition 7.8.1) such that the image $I' \subset I$ of $\alpha$ is finite. By Homology, Lemma 12.5.8 it suffices to show that for any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ turns the sequence of the lemma into an exact sequence. By (18.19.2.1) we obtain the usual sequence

$0 \to \mathcal{F}(U) \to \prod \nolimits _{i \in I'} \mathcal{F}(U_ i) \to \prod \nolimits _{i, i' \in I'} \mathcal{F}(U_ i \times _ U U_{i'})$

This is an exact sequence by Sites, Lemma 7.8.6 applied to the family of maps $\{ U_ i \to U\} _{i \in I'}$ which is refined by the covering $\mathcal{V}$. $\square$

Lemma 18.30.3. 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 7.17.7. 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 7.17.7. The finite case is dealt with in Lemma 18.3.2 (it holds over any object of $\mathcal{C}$). $\square$

Lemma 18.30.4. 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 $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{O}_ U, \mathcal{F}) = \mathcal{F}(U)$ by (18.19.2.1) and because the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ commutes with direct sums by Lemma 18.30.3. $\square$

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

Situation 18.30.5. 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 (Sites, Section 7.17).

3. For a 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 18.30.6. In Situation 18.30.5 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 $\mathcal{B}$.

Lemma 18.30.7. In Situation 18.30.5 assume (1) and (2) hold.

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

18.30.7.1
$$\label{sites-modules-equation-towards-constructible-sets} \text{Coequalizer}\left( \xymatrix{ \coprod \nolimits _{j = 1, \ldots , m} h_{V_ j}^\# \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{i = 1, \ldots , n} h_{U_ i}^\# } \right)$$

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

18.30.7.2
$$\label{sites-modules-equation-towards-constructible} \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{j = 1, \ldots , m} j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i} \right)$$

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

Proof. Proof of (1). By Lemma 18.30.6 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 Sites, Lemma 7.17.7. (Details omitted; hint: an infinite coproduct is the filtered colimit of the finite sub-coproducts.) Thus our sheaf is the colimit of the cokernels of these maps between finite coproducts.

Proof of (2). By Lemma 18.30.6 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 18.30.4. 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 (18.30.7.2). 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 $s \in \mathcal{F}(W)$. Since $\bigoplus j_{U_ i!}\mathcal{O}_{U_ i} \to \mathcal{F}$ is surjective, by (1) we may choose a covering $\{ W_ k \to W\} _{k \in K}$ with $W_ k \in \mathcal{B}$ such that $s|_{W_ k}$ is the image of some section $s_ k$ of $\bigoplus j_{U_ i!}\mathcal{O}_{U_ i})$. By (2) the object $W$ is quasi-compact. By Lemma 18.30.2 there is a finite subset $K' \subset K$ such that $\bigoplus _{k \in K'} j_{W_ k!}\mathcal{O}_{W_ k} \to j_{W!}\mathcal{O}_ W$ is surjective. We conclude that $\mathop{\mathrm{Coker}}(\varphi )$ is equal to

$\mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{k \in K'} j_{W_ k!}\mathcal{O}_{W_ k} \oplus \bigoplus j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus j_{U_ i!}\mathcal{O}_{U_ i} \right)$

where the map $\bigoplus _{k \in K'} j_{W_ k!}\mathcal{O}_{W_ k} \to \bigoplus j_{U_ i!}\mathcal{O}_{U_ i}$ corresponds to $\sum _{k \in K'} s_ k$. This finishes the proof. $\square$

Lemma 18.30.9. In Situation 18.30.5 assume (1), (2), and (3) hold. Let $\mathcal{O}$ be a sheaf of rings. Assume 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 coverings $\{ U_{ik} \to U_ i\} _{k \in K_ i}$ with $U_{ik} \in \mathcal{B}$. Then there exist finite subsets $K'_ i \subset K_ i$ and a finite set $L$ of $W_ l \in \mathcal{B}$ and a commutative diagram

$\xymatrix{ \bigoplus _{l \in L} j_{W_ l!}\mathcal{O}_{W_ l} \ar[d] \ar[r] & \bigoplus _{i = 1, \ldots , n} \bigoplus _{k \in K'_ i} j_{U_{ik}!}\mathcal{O}_{U_{ik}} \ar[d] \\ \bigoplus _{j = 1, \ldots , m} j_{V_ j!}\mathcal{O}_{V_ j} \ar[r] & \bigoplus _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i} }$

inducing an isomorphism on cokernels of the horizontal maps.

Proof. Since $U_ i$ is quasi-compact, we may choose finite subsets $K'_ i \subset K_ i$ as in Lemma 18.30.2. Then since $\bigoplus _{i = 1, \ldots , n} \bigoplus _{k \in K'_ i} j_{U_{ik}!}\mathcal{O}_{U_{ik}} \to \bigoplus _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i}$ is surjective, we can find coverings $\{ V_{jm} \to V_ j\} _{m \in M_ j}$ with $V_{jm} \in \mathcal{B}$ such that we can find a commutative diagram

$\xymatrix{ \bigoplus _{j = 1, \ldots , m} \bigoplus _{m \in M_ j} j_{V_{jm}!}\mathcal{O}_{V_{jm}} \ar[d] \ar[r] & \bigoplus _{i = 1, \ldots n} \bigoplus _{k \in K'_ i} j_{U_{ik}!}\mathcal{O}_{U_{ik}} \ar[d] \\ \bigoplus _{j = 1, \ldots , m} j_{V_ j!}\mathcal{O}_{V_ j} \ar[r] & \bigoplus _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i} }$

Since $V_ j$ is quasi-compact, we can choose finite subsets $M'_ j \subset M_ j$ as in Lemma 18.30.2. Set

$L = \left(\coprod \nolimits _{i = 1, \ldots , n} K'_ i \times K'_ i \right) \coprod \left(\coprod \nolimits _{j = 1, \ldots , m} M'_ j\right)$

and for $l = (k, k') \in K'_ i \times K'_ i \subset L$ set $W_ l = U_{ik} \times _{U_ i} U_{ik'}$ and for $l = m \in M'_ j \subset L$ set $W_ l = V_{jm}$. Since we have the exact sequences of Lemma 18.30.2 for the families $\{ U_{ik} \to U_ i\} _{k \in K'_ i}$ we conclude that we get a diagram as in the statement of the lemma (details omitted), except that it is not yet clear that $W_ l \in \mathcal{B}$. However, since $W_ l$ is quasi-compact for all $l \in L$ we do another application of Lemma 18.30.2 and find finite families of maps $\{ W_{lt} \to W_ l\} _{t \in T_ l}$ with $W_{lt} \in \mathcal{B}$ such that $\bigoplus j_{W_{lt}!}\mathcal{O}_{W_{lt}} \to j_{W_ l!}\mathcal{O}_{W_ l}$ is surjective. Then we replace $L$ by $\coprod _{l \in L} T_ l$ and everything is clear. $\square$

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 (18.30.7.2). 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 $U \in \mathcal{B}$. Since $\mathcal{F}_2 \to \mathcal{F}_3$ is surjective, we can choose coverings $\{ W_{ik} \to W_ i\}$ with $W_{ik} \in \mathcal{B}$ such that $A_{W_{ik}} \to \mathcal{F}_3$ lifts to a map $A_{W_{ik}} \to \mathcal{F}_2$. By Lemma 18.30.9 we may replace our collection $\{ W_ i\}$ by a finite subcollection of the collection $\{ W_{ik}\}$ and 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 surjects onto $\mathcal{K}_3 = \mathop{\mathrm{Ker}}(\bigoplus A_{W_ i} \to \mathcal{F}_3)$. Thus, arguing as above, after replacing each $T_ j$ by a finite family of elements of $\mathcal{B}$ (permissible by Lemma 18.30.2) 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 18.30.11. In Situation 18.30.5 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 (18.30.7.2). 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 12.10.3. By the results of Lemmas 18.30.8 and 18.30.10 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 18.30.1 and 18.30.9 and arguing as in the proof of Lemma 18.30.10 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 18.30.8. $\square$

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