Lemma 66.21.1. Let S be a scheme. Let X be an algebraic space over S. The rule U \mapsto \Gamma (U, \mathcal{O}_ U) defines a sheaf of rings on X_{\acute{e}tale}.
66.21 The structure sheaf of an algebraic space
The structure sheaf of an algebraic space is the sheaf of rings of the following lemma.
Proof. Immediate from the definition of a covering and Descent, Lemma 35.8.1. \square
Definition 66.21.2. Let S be a scheme. Let X be an algebraic space over S. The structure sheaf of X is the sheaf of rings \mathcal{O}_ X on the small étale site X_{\acute{e}tale} described in Lemma 66.21.1.
According to Lemma 66.18.13 the sheaf \mathcal{O}_ X corresponds to a system of étale sheaves (\mathcal{O}_ X)_ U for U ranging through the objects of X_{\acute{e}tale}. It is clear from the proof of that lemma and our definition that we have simply (\mathcal{O}_ X)_ U = \mathcal{O}_ U where \mathcal{O}_ U is the structure sheaf of U_{\acute{e}tale} as introduced in Descent, Definition 35.8.2. In particular, if X is a scheme we recover the sheaf \mathcal{O}_ X on the small étale site of X.
Via the equivalence \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) = \mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}}) of Lemma 66.18.3 we may also think of \mathcal{O}_ X as a sheaf of rings on X_{spaces, {\acute{e}tale}}. It is explained in Remark 66.18.4 how to compute \mathcal{O}_ X(Y), and in particular \mathcal{O}_ X(X), when Y \to X is an object of X_{spaces, {\acute{e}tale}}.
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,
The construction f \mapsto (f_{small}, f^\sharp ) is compatible with compositions.
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
Lemma 66.21.4. Let S be a scheme. Let X be an algebraic space over S. The following are equivalent
X is reduced,
for every x \in |X| the local ring of X at x is reduced (Remark 66.7.6).
In this case \Gamma (X, \mathcal{O}_ X) is a reduced ring and if f \in \Gamma (X, \mathcal{O}_ X) has X = V(f), then f = 0.
Proof. The equivalence of (1) and (2) follows from Properties, Lemma 28.3.2 applied to affine schemes étale over X. The final statements follow the cited lemma and fact that \Gamma (X, \mathcal{O}_ X) is a subring of \Gamma (U, \mathcal{O}_ U) for some reduced scheme U étale over X. \square
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