The Stacks project

Lemma 34.8.15. Let $\mathcal{F}$ be a presheaf on $(\mathit{Sch}/S)_{ph}$. Then $\mathcal{F}$ is a sheaf if and only if

  1. $\mathcal{F}$ satisfies the sheaf condition for Zariski coverings, and

  2. if $f : V \to U$ is proper surjective, then $\mathcal{F}(U)$ maps bijectively to the equalizer of the two maps $\mathcal{F}(V) \to \mathcal{F}(V \times _ U V)$.

Moreover, in the presence of (1) property (2) is equivalent to property

  1. the sheaf property for $\{ V \to U\} $ as in (2) with $U$ affine.

Proof. We will show that if (1) and (2) hold, then $\mathcal{F}$ is sheaf. Let $\{ T_ i \to T\} $ be a ph covering, i.e., a covering in $(\mathit{Sch}/S)_{ph}$. We will verify the sheaf condition for this covering. Let $s_ i \in \mathcal{F}(T_ i)$ be sections which restrict to the same section over $T_ i \times _ T T_{i'}$. We will show that there exists a unique section $s \in \mathcal{F}(T)$ restricting to $s_ i$ over $T_ i$. Let $T = \bigcup U_ j$ be an affine open covering. By property (1) it suffices to produce sections $s_ j \in \mathcal{F}(U_ j)$ which agree on $U_ j \cap U_{j'}$ in order to produce $s$. Consider the ph coverings $\{ T_ i \times _ T U_ j \to U_ j\} $. Then $s_{ji} = s_ i|_{T_ i \times _ T U_ j}$ are sections agreeing over $(T_ i \times _ T U_ j) \times _{U_ j} (T_{i'} \times _ T U_ j)$. Choose a proper surjective morphism $V_ j \to U_ j$ and a finite affine open covering $V_ j = \bigcup V_{jk}$ such that the standard ph covering $\{ V_{jk} \to U_ j\} $ refines $\{ T_ i \times _ T U_ j \to U_ j\} $. If $s_{jk} \in \mathcal{F}(V_{jk})$ denotes the pullback of $s_{ji}$ to $V_{jk}$ by the implied morphisms, then we find that $s_{jk}$ glue to a section $s'_ j \in \mathcal{F}(V_ j)$. Using the agreement on overlaps once more, we find that $s'_ j$ is in the equalizer of the two maps $\mathcal{F}(V_ j) \to \mathcal{F}(V_ j \times _{U_ j} V_ j)$. Hence by (2) we find that $s'_ j$ comes from a unique section $s_ j \in \mathcal{F}(U_ j)$. We omit the verification that these sections $s_ j$ have all the desired properties.

Proof of the equivalence of (2) and (2') in the presence of (1). Suppose $V \to U$ is a morphism of $(\mathit{Sch}/S)_{ph}$ which is proper and surjective. Choose an affine open covering $U = \bigcup U_ i$ and set $V_ i = V \times _ U U_ i$. Then we see that $\mathcal{F}(U) \to \mathcal{F}(V)$ is injective because we know $\mathcal{F}(U_ i) \to \mathcal{F}(V_ i)$ is injective by (2') and we know $\mathcal{F}(U) \to \prod \mathcal{F}(U_ i)$ is injective by (1). Finally, suppose that we are given an $t \in \mathcal{F}(V)$ in the equalizer of the two maps $\mathcal{F}(V) \to \mathcal{F}(V \times _ U V)$. Then $t|_{V_ i}$ is in the equalizer of the two maps $\mathcal{F}(V_ i) \to \mathcal{F}(V_ i \times _{U_ i} V_ i)$ for all $i$. Hence we obtain a unique section $s_ i \in \mathcal{F}(U_ i)$ mapping to $t|_{V_ i}$ for all $i$ by (2'). We omit the verification that $s_ i|_{U_ i \cap U_ j} = s_ j|_{U_ i \cap U_ j}$ for all $i, j$; this uses the uniqueness property just shown. By the sheaf property for the covering $U = \bigcup U_ i$ we obtain a section $s \in \mathcal{F}(U)$. We omit the proof that $s$ maps to $t$ in $\mathcal{F}(V)$. $\square$

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