nLab
relative scheme

Sch/SSch/S is the slice category of schemes over a fixed scheme SS. The schemes over SS are morphisms of schemes f:RSf: R\to S, called also relative schemes over SS or SS-schemes. A morphism g:(f:RS)(f:RS)g: (f:R\to S)\to (f':R'\to S') is a morphism g:RRg:R\to R' of schemes such that fg=ff'\circ g = f; morphisms of SS-schemes are also called morphisms of schemes over SS, and pictured by commutative triangles.

Every scheme can be considered a Spec()Spec(\mathbb{Z})-scheme. For any (commutative unital) ring kk, a kk-scheme is a synonym for Spec(k)Spec(k)-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 f:RSf:R\to S of schemes is quasicompact (or RR is quasicompact as an SS-scheme) if the preimage f 1(U)f^{-1}(U) of any affine USU\subset S 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:

  • Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64. Springer 1972. vi+160 pp.

Revised on March 6, 2013 19:27:31 by Zoran Škoda (161.53.130.104)