## 39.2 Notation

Let $S$ be a scheme. If $U$, $T$ are schemes over $S$ we denote $U(T)$ for the set of $T$-valued points of $U$ over $S$. In a formula: $U(T) = \mathop{\mathrm{Mor}}\nolimits _ S(T, U)$. We try to reserve the letter $T$ to denote a “test scheme” over $S$, as in the discussion that follows. Suppose we are given schemes $X$, $Y$ over $S$ and a morphism of schemes $f : X \to Y$ over $S$. For any scheme $T$ over $S$ we get an induced map of sets

$f : X(T) \longrightarrow Y(T)$

which as indicated we denote by $f$ also. In fact this construction is functorial in the scheme $T/S$. Yoneda's Lemma, see Categories, Lemma 4.3.5, says that $f$ determines and is determined by this transformation of functors $f : h_ X \to h_ Y$. More generally, we use the same notation for maps between fibre products. For example, if $X$, $Y$, $Z$ are schemes over $S$, and if $m : X \times _ S Y \to Z \times _ S Z$ is a morphism of schemes over $S$, 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.

We continue our convention to label projection maps starting with index $0$, so we have $\text{pr}_0 : X \times _ S Y \to X$ and $\text{pr}_1 : X \times _ S Y \to Y$.

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