The Stacks project

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. $\square$

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

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 $\mathcal{V}$. $\square$

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 $\mathcal{C}$). $\square$

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. $\square$

Proof. Part (1) follows from Sites, Lemmas 7.12.5 and 7.12.4. Part (2) follows from Lemmas 18.28.8 and 18.30.1. $\square$

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

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

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

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. $\square$

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

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

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

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

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. $\square$

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

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

Then we see that

and the lemma follows from the assumption and Lemma 18.30.8. $\square$