Definition 26.18.1. Let $S$ be a scheme.

We say $X$ is a

*scheme over $S$*to mean that $X$ comes equipped with a morphism of schemes $X \to S$. The morphism $X \to S$ is sometimes called the*structure morphism*.If $R$ is a ring we say $X$ is a

*scheme over $R$*instead of $X$ is a scheme over $\mathop{\mathrm{Spec}}(R)$.A

*morphism $f : X \to Y$ of schemes over $S$*is a morphism of schemes such that the composition $X \to Y \to S$ of $f$ with the structure morphism of $Y$ is equal to the structure morphism of $X$.We denote $\mathop{\mathrm{Mor}}\nolimits _ S(X, Y)$ the set of all morphisms from $X$ to $Y$ over $S$.

Let $X$ be a scheme over $S$. Let $S' \to S$ be a morphism of schemes. The

*base change*of $X$ is the scheme $X_{S'} = S' \times _ S X$ over $S'$.Let $f : X \to Y$ be a morphism of schemes over $S$. Let $S' \to S$ be a morphism of schemes. The

*base change*of $f$ is the induced morphism $f' : X_{S'} \to Y_{S'}$ (namely the morphism $\text{id}_{S'} \times _{\text{id}_ S} f$).Let $R$ be a ring. Let $X$ be a scheme over $R$. Let $R \to R'$ be a ring map. The

*base change*$X_{R'}$ is the scheme $\mathop{\mathrm{Spec}}(R') \times _{\mathop{\mathrm{Spec}}(R)} X$ over $R'$.

## Comments (1)

Comment #207 by Rex on

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