The Stacks project

Lemma 21.16.5. Let $\mathcal{I}$ be a cofiltered index category and let $(\mathcal{C}_ i, f_ a)$ be an inverse system of sites over $\mathcal{I}$ as in Sites, Situation 7.18.1. Set $\mathcal{C} = \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i$ as in Sites, Lemmas 7.18.2 and 7.18.3. Moreover, assume given

  1. an abelian sheaf $\mathcal{F}_ i$ on $\mathcal{C}_ i$ for all $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$,

  2. for $a : j \to i$ a map $\varphi _ a : f_ a^{-1}\mathcal{F}_ i \to \mathcal{F}_ j$ of abelian sheaves on $\mathcal{C}_ j$

such that $\varphi _ c = \varphi _ b \circ f_ b^{-1}\varphi _ a$ whenever $c = a \circ b$. Then there exists a map of systems $(\mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ such that $\mathcal{F}_ i \to \mathcal{G}_ i$ is injective and $\mathcal{G}_ i$ is an injective abelian sheaf.

Proof. For each $i$ we pick an injection $\mathcal{F}_ i \to \mathcal{A}_ i$ where $\mathcal{A}_ i$ is an injective abelian sheaf on $\mathcal{C}_ i$. Then we can consider the family of maps

\[ \gamma _ i : \mathcal{F}_ i \longrightarrow \prod \nolimits _{b : k \to i} f_{b, *}\mathcal{A}_ k = \mathcal{G}_ i \]

where the component maps are the maps adjoint to the maps $f_ b^{-1}\mathcal{F}_ i \to \mathcal{F}_ k \to \mathcal{A}_ k$. For $a : j \to i$ in $\mathcal{I}$ there is a canonical map

\[ \psi _ a : f_ a^{-1}\mathcal{G}_ i \to \mathcal{G}_ j \]

whose components are the canonical maps $f_ b^{-1}f_{a \circ b, *}\mathcal{A}_ k \to f_{b, *}\mathcal{A}_ k$ for $b : k \to j$. Thus we find an injection $(\gamma _ i) : (\mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ of systems of abelian sheaves. Note that $\mathcal{G}_ i$ is an injective sheaf of abelian groups on $\mathcal{C}_ i$, see Lemma 21.14.2 and Homology, Lemma 12.27.3. This finishes the construction. $\square$

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