The Stacks project

Lemma 7.17.10. Let $\mathcal{C}$ be a site. Let $\beta $ be an ordinal. Let $\beta \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, $\alpha \mapsto \mathcal{F}_\alpha $ be a system of sheaves over $\beta $. For $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ consider the canonical map

\[ \mathop{\mathrm{colim}}\nolimits _{\alpha < \beta } \mathcal{F}_\alpha (U) \longrightarrow \left(\mathop{\mathrm{colim}}\nolimits _{\alpha < \beta } \mathcal{F}_\alpha \right)(U) \]

If the cofinality of $\beta $ is large enough, then this map is bijective for all $U$.

Proof. The left hand side is the value on $U$ of the colimit $\mathcal{F}_{\mathop{\mathrm{colim}}\nolimits }$ taken in the category of presheaves, see Section 7.4. Recall that $\mathop{\mathrm{colim}}\nolimits _{\alpha < \beta } \mathcal{F}_\alpha $ is the sheafification $\mathcal{F}_{\mathop{\mathrm{colim}}\nolimits }^\# $ of $\mathcal{F}_{\mathop{\mathrm{colim}}\nolimits }$, see Lemma 7.10.13. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be an element of the set $\text{Cov}(\mathcal{C})$ of coverings of $\mathcal{C}$. If the cofinality of $\beta $ is larger than the cardinality of $I$, then we claim

\[ H^0(\mathcal{U}, \mathcal{F}_{\mathop{\mathrm{colim}}\nolimits }) = \mathop{\mathrm{colim}}\nolimits H^0(\mathcal{U}, \mathcal{F}_\alpha ) = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_\alpha (U) = \mathcal{F}_{\mathop{\mathrm{colim}}\nolimits }(U) \]

The second and third equality signs are clear. For the first, say $s = (s_ i) \in H^0(\mathcal{U}, \mathcal{F}_{\mathop{\mathrm{colim}}\nolimits })$. Then for each $i$ the element $s_ i$ comes from an element $s_{i, \alpha _ i} \in \mathcal{F}_{\alpha _ i}(U_ i)$ for some $\alpha _ i < \beta $. By the assumption on cofinality, we can choose $\alpha _ i = \alpha $ independent of $i$. Then $s_ i$ and $s_ j$ map to the same element of $\mathcal{F}_{\alpha _{i, j}}(U_ i \times _ U U_ j)$ for some $\alpha _{i, j} < \beta $. Since the cardinality if $I \times I$ is also less than the cofinality of $\beta $, we see that we may after increasing $\alpha $ assume $\alpha _{i, j} = \alpha $ for all $i, j$. This proves that the natural map $\mathop{\mathrm{colim}}\nolimits H^0(\mathcal{U}, \mathcal{F}_\alpha ) \to H^0(\mathcal{U}, \mathcal{F}_{\mathop{\mathrm{colim}}\nolimits })$ is surjective. A very similar argument shows that it is injective. In particular, we see that $\mathcal{F}_{\mathop{\mathrm{colim}}\nolimits }$ satisfies the sheaf condition for $\mathcal{U}$. Thus if the cofinality of $\beta $ is larger than the supremum of the cardinalities of the set of index sets $I$ of coverings, then we conclude. $\square$

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