The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Definition 25.18.1. Let $S$ be a scheme.

  1. 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.

  2. If $R$ is a ring we say $X$ is a scheme over $R$ instead of $X$ is a scheme over $\mathop{\mathrm{Spec}}(R)$.

  3. 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 $X$.

  4. We denote $\mathop{Mor}\nolimits _ S(X, Y)$ the set of all morphisms from $X$ to $Y$ over $S$.

  5. 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 $S'$.

  6. 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 $\text{id}_{S'} \times _{\text{id}_ S} f$).

  7. 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

Typo: "namely the morphsm"

There are also:

  • 8 comment(s) on Section 25.18: Base change in algebraic geometry

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