The Stacks project

Lemma 58.15.6. Let $\mathcal{F}$ be a presheaf on $\mathit{Sch}/S$. Then $\mathcal{F}$ satisfies the sheaf property for the fpqc topology if and only if

  1. $\mathcal{F}$ satisfies the sheaf property with respect to the Zariski topology, and

  2. for every faithfully flat morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ of affine schemes over $S$, the sheaf axiom holds for the covering $\{ \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)\} $. Namely, this means that

    \[ \xymatrix{ \mathcal{F}(\mathop{\mathrm{Spec}}(A)) \ar[r] & \mathcal{F}(\mathop{\mathrm{Spec}}(B)) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(\mathop{\mathrm{Spec}}(B \otimes _ A B)) } \]

    is an equalizer diagram.

Proof. See Topologies, Lemma 34.9.13. $\square$


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