Morphism of schemes

In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.

More generally, morphisms p:X →S with various schemes X but fixed scheme S form the category of schemes over S (the slice category of the category of schemes with the base object S.) An object in the category is called an S-scheme and a morphism in the category an S-morphism; explicitly, an S-morphism from p:X →S to q:Y →S is a morphism ƒ:X →Y of schemes such that p = q ∘ ƒ.