# fibre product

Let $S$ be a scheme, and let $i:X\longrightarrow S$ and $j:Y\longrightarrow S$ be schemes over $S$. A fibre product of $X$ and $Y$ over $S$ is a scheme $X\times_{S}Y$ together with morphisms

 $\displaystyle p:X\times_{S}Y\longrightarrow X$ $\displaystyle q:X\times_{S}Y\longrightarrow Y$

such that given any scheme $Z$ with morphisms

 $\displaystyle x:Z\longrightarrow X$ $\displaystyle y:Z\longrightarrow Y$

where $i\circ x=j\circ y$, there exists a unique morphism

 $(x,y):Z\longrightarrow X\times_{S}Y$

making the diagram