The Stacks project

Lemma 66.21.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then there is a canonical map $f^\sharp : f_{small}^{-1}\mathcal{O}_ Y \to \mathcal{O}_ X$ such that

\[ (f_{small}, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y) \]

is a morphism of ringed topoi. Furthermore,

  1. The construction $f \mapsto (f_{small}, f^\sharp )$ is compatible with compositions.

  2. If $f$ is a morphism of schemes, then $f^\sharp $ is the map described in Descent, Remark 35.8.4.

Proof. By Lemma 66.18.10 it suffices to give an $f$-map from $\mathcal{O}_ Y$ to $\mathcal{O}_ X$. In other words, for every commutative diagram

\[ \xymatrix{ U \ar[d]_ g \ar[r] & X \ar[d]^ f \\ V \ar[r] & Y } \]

where $U \in X_{\acute{e}tale}$, $V \in Y_{\acute{e}tale}$ we have to give a map of rings $ (f^\sharp )_{(U, V, g)} : \Gamma (V, \mathcal{O}_ V) \to \Gamma (U, \mathcal{O}_ U). $ Of course we just take $(f^\sharp )_{(U, V, g)} = g^\sharp $. It is clear that this is compatible with restriction mappings and hence indeed gives an $f$-map. We omit checking compatibility with compositions and agreement with the construction in Descent, Remark 35.8.4. $\square$

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