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The Stacks project

Theorem 59.40.5. Let X, Y be schemes. Let

(g, g^\# ) : (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y)

be a morphism of locally ringed topoi. Then there exists a unique morphism of schemes f : X \to Y such that (g, g^\# ) is isomorphic to (f_{small}, f_{small}^\sharp ). In other words, the construction

\mathit{Sch}\longrightarrow \textit{Locally ringed topoi}, \quad X \longrightarrow (X_{\acute{e}tale}, \mathcal{O}_ X)

is fully faithful (morphisms up to 2-isomorphisms on the right hand side).

Proof. You can prove this theorem by carefully adjusting the arguments of the proof of Lemma 59.40.4 to the global setting. However, we want to indicate how we can glue the result of that lemma to get a global morphism due to the rigidity provided by the result of Lemma 59.40.2. Unfortunately, this is a bit messy.

Let us prove existence when Y is affine. In this case choose an affine open covering X = \bigcup U_ i. For each i the inclusion morphism j_ i : U_ i \to X induces a morphism of locally ringed topoi (j_{i, small}, j_{i, small}^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (U_{i, {\acute{e}tale}}), \mathcal{O}_{U_ i}) \to (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) by Lemma 59.40.1. We can compose this with (g, g^\sharp ) to obtain a morphism of locally ringed topoi

(g, g^\sharp ) \circ (j_{i, small}, j_{i, small}^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (U_{i, {\acute{e}tale}}), \mathcal{O}_{U_ i}) \to (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y)

see Modules on Sites, Lemma 18.40.10. By Lemma 59.40.4 there exists a unique morphism of schemes f_ i : U_ i \to Y and a 2-isomorphism

t_ i : (f_{i, small}, f_{i, small}^\sharp ) \longrightarrow (g, g^\sharp ) \circ (j_{i, small}, j_{i, small}^\sharp ).

Set U_{i, i'} = U_ i \cap U_{i'}, and denote j_{i, i'} : U_{i, i'} \to U_ i the inclusion morphism. Since we have j_ i \circ j_{i, i'} = j_{i'} \circ j_{i', i} we see that

\begin{align*} (g, g^\sharp ) \circ (j_{i, small}, j_{i, small}^\sharp ) \circ (j_{i, i', small}, j_{i, i', small}^\sharp ) = \\ (g, g^\sharp ) \circ (j_{i', small}, j_{i', small}^\sharp ) \circ (j_{i', i, small}, j_{i', i, small}^\sharp ) \end{align*}

Hence by uniqueness (see Lemma 59.40.3) we conclude that f_ i \circ j_{i, i'} = f_{i'} \circ j_{i', i}, in other words the morphisms of schemes f_ i = f \circ j_ i are the restrictions of a global morphism of schemes f : X \to Y. Consider the diagram of 2-isomorphisms (where we drop the components {}^\sharp to ease the notation)

\xymatrix{ g \circ j_{i, small} \circ j_{i, i', small} \ar[rr]^{t_ i \star \text{id}_{j_{i, i', small}}} \ar@{=}[d] & & f_{small} \circ j_{i, small} \circ j_{i, i', small} \ar@{=}[d] \\ g \circ j_{i', small} \circ j_{i', i, small} \ar[rr]^{t_{i'} \star \text{id}_{j_{i', i, small}}} & & f_{small} \circ j_{i', small} \circ j_{i', i, small} }

The notation \star indicates horizontal composition, see Categories, Definition 4.29.1 in general and Sites, Section 7.36 for our particular case. By the result of Lemma 59.40.2 this diagram commutes. Hence for any sheaf \mathcal{G} on Y_{\acute{e}tale} the isomorphisms t_ i : f_{small}^{-1}\mathcal{G}|_{U_ i} \to g^{-1}\mathcal{G}|_{U_ i} agree over U_{i, i'} and we obtain a global isomorphism t : f_{small}^{-1}\mathcal{G} \to g^{-1}\mathcal{G}. It is clear that this isomorphism is functorial in \mathcal{G} and is compatible with the maps f_{small}^\sharp and g^\sharp (because it is compatible with these maps locally). This proves the theorem in case Y is affine.

In the general case, let V \subset Y be an affine open. Then h_ V is a subsheaf of the final sheaf * on Y_{\acute{e}tale}. As g is exact we see that g^{-1}h_ V is a subsheaf of the final sheaf on X_{\acute{e}tale}. Hence by Lemma 59.31.1 there exists an open subscheme W \subset X such that g^{-1}h_ V = h_ W. By Modules on Sites, Lemma 18.40.12 there exists a commutative diagram of morphisms of locally ringed topoi

\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (W_{\acute{e}tale}), \mathcal{O}_ W) \ar[r] \ar[d]_{g'} & (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \ar[d]^ g \\ (\mathop{\mathit{Sh}}\nolimits (V_{\acute{e}tale}), \mathcal{O}_ V) \ar[r] & (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y) }

where the horizontal arrows are the localization morphisms (induced by the inclusion morphisms V \to Y and W \to X) and where g' is induced from g. By the result of the preceding paragraph we obtain a morphism of schemes f' : W \to V and a 2-isomorphism t : (f'_{small}, (f'_{small})^\sharp ) \to (g', (g')^\sharp ). Exactly as before these morphisms f' (for varying affine opens V \subset Y) agree on overlaps by uniqueness, so we get a morphism f : X \to Y. Moreover, the 2-isomorphisms t are compatible on overlaps by Lemma 59.40.2 again and we obtain a global 2-isomorphism (f_{small}, (f_{small})^\sharp ) \to (g, (g)^\sharp ). as desired. Some details omitted. \square


Comments (2)

Comment #5017 by Matthieu Romagny on

Typo: in the second paragraph of the proof, after we compose with (g,g♯) we obtain a morphism of locally ringed topoi whose target is (Sh(Y_étale),O_Y), not (Sh(X_étale),O_X).


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