The Stacks project

Lemma 18.40.8. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Consider the following conditions

  1. The diagram of sheaves

    \[ \xymatrix{ f^{-1}(\mathcal{O}^*_\mathcal {D}) \ar[r]_-{f^\sharp } \ar[d] & \mathcal{O}^*_\mathcal {C} \ar[d] \\ f^{-1}(\mathcal{O}_\mathcal {D}) \ar[r]^-{f^\sharp } & \mathcal{O}_\mathcal {C} } \]

    is cartesian.

  2. For any point $p$ of $\mathcal{C}$, setting $q = f \circ p$, the diagram

    \[ \xymatrix{ \mathcal{O}^*_{\mathcal{D}, q} \ar[r] \ar[d] & \mathcal{O}^*_{\mathcal{C}, p} \ar[d] \\ \mathcal{O}_{\mathcal{D}, q} \ar[r] & \mathcal{O}_{\mathcal{C}, p} } \]

    of sets is cartesian.

We always have (1) $\Rightarrow $ (2). If $\mathcal{C}$ has enough points then (1) and (2) are equivalent. If $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ and $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ are locally ringed topoi then (2) is equivalent to

  1. For any point $p$ of $\mathcal{C}$, setting $q = f \circ p$, the ring map $\mathcal{O}_{\mathcal{D}, q} \to \mathcal{O}_{\mathcal{C}, p}$ is a local ring map.

In fact, properties (2), or (3) for a conservative family of points implies (1).

Proof. This lemma proves itself, in other words, it follows by unwinding the definitions. $\square$

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