The Stacks project

76.3 Notation

Let $S$ be a scheme; this will be our base scheme and all algebraic spaces will be over $S$. Let $B$ be an algebraic space over $S$; this will be our base algebraic space, and often other algebraic spaces, and schemes will be over $B$. If we say that $X$ is an algebraic space over $B$, then we mean that $X$ is an algebraic space over $S$ which comes equipped with structure morphism $X \to B$. Moreover, we try to reserve the letter $T$ to denote a “test” scheme over $B$. In other words $T$ is a scheme which comes equipped with a structure morphism $T \to B$. In this situation we denote $X(T)$ for the set of $T$-valued points of $X$ over $B$. In a formula:

\[ X(T) = \mathop{Mor}\nolimits _ B(T, X). \]

Similarly, given a second algebraic space $Y$ over $B$ we set

\[ X(Y) = \mathop{Mor}\nolimits _ B(Y, X). \]

Suppose we are given algebraic spaces $X$, $Y$ over $B$ as above and a morphism $f : X \to Y$ over $B$. For any scheme $T$ over $B$ we get an induced map of sets

\[ f : X(T) \longrightarrow Y(T) \]

which is functorial in the scheme $T$ over $B$. As $f$ is a map of sheaves on $(\mathit{Sch}/S)_{fppf}$ over the sheaf $B$ it is clear that $f$ determines and is determined by this rule. More generally, we use the same notation for maps between fibre products. For example, if $X$, $Y$, $Z$ are algebraic spaces over $B$, and if $m : X \times _ B Y \to Z \times _ B Z$ is a morphism of algebraic spaces over $B$, then we think of $m$ as corresponding to a collection of maps between $T$-valued points

\[ X(T) \times Y(T) \longrightarrow Z(T) \times Z(T). \]

And so on and so forth.

Finally, given two maps $f, g : X \to Y$ of algebraic spaces over $B$, if the induced maps $f, g : X(T) \to Y(T)$ are equal for every scheme $T$ over $B$, then $f = g$, and hence also $f, g : X(Z) \to Y(Z)$ are equal for every third algebraic space $Z$ over $B$. Hence, for example, to check the axioms for an group algebraic space $G$ over $B$, it suffices to check commutativity of diagram on $T$-valued points where $T$ is a scheme over $B$ as we do in Definition 76.5.1 below.


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