Scheme over a base - Knowlpedia

Let $S$ be a scheme. A scheme over $S$ , or $S$ -scheme, is a scheme $X$ equipped with a specified morphism of schemes

X\longrightarrow S,

called its structure morphism.

Given $S$ -schemes $X\to S$ and $Y\to S$ , an $S$ -morphism $f:X\to Y$ is a morphism for which the composite $X\to Y\to S$ equals the specified map $X\to S$ . Schemes over $S$ , together with their $S$ -morphisms, form the category customarily denoted $\mathbf{Sch}/S$ .

Examples

For example, a ring homomorphism $R\to A$ makes $\operatorname{Spec}A$ a scheme over $\operatorname{Spec}R$ . Constructions such as base change and the fiber product keep track of these specified maps to the base.