is the slice category of schemes over a fixed scheme . The schemes over are morphisms of schemes , called also relative schemes over or -schemes. A morphism is a morphism of schemes such that ; morphisms of -schemes are also called morphisms of schemes over , and pictured by commutative triangles.
Every scheme can be considered a -scheme. For any (commutative unital) ring , a -scheme is a synonym for -scheme.
Grothendieck has emphasised the relative point of view: the emphasis of the basic theory of schemes should not be on the properties of schemes, but on the properties of morphisms.
Many definitions of local properties of schemes, can be automatically generalized to morphisms, by looking at properties of preimages of the affine covers of the base scheme. For example, a morphism of schemes is quasicompact (or is quasicompact as an -scheme) if the preimage of any affine is a quasicompact scheme. One can also talk about projective?, affine, quasiprojective?, proper, separated etc. morphisms.
Relative schemes over a general ringed topos are developed in a thesis under Grothendieck’s guidance: