The Stacks project

Definition 26.18.1. 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 $\mathop{\mathrm{Mor}}\nolimits _ S(X, Y)$ 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:

  • 12 comment(s) on Section 26.18: Base change in algebraic geometry

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01JX. Beware of the difference between the letter 'O' and the digit '0'.