The Stacks project

Lemma 21.9.2. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be an abelian presheaf on $\mathcal{C}$. The following are equivalent

  1. $\mathcal{F}$ is an abelian sheaf on $\mathcal{C}$ and

  2. for every covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ of the site $\mathcal{C}$ the natural map

    \[ \mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F}) \]

    (see Sites, Section 7.10) is bijective.

Proof. This is true since the sheaf condition is exactly that $\mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F})$ is bijective for every covering of $\mathcal{C}$. $\square$


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