The Stacks project

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.

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 $X_{\acute{e}tale}$.

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,

  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. $\square$

Lemma 66.21.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent

  1. $X$ is reduced,

  2. 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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