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functorial geometry

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Idea

In algebraic geometry, there are two equivalent ways of looking at a scheme: it may be viewed

  1. as a petit topos with a structure sheaf of commutative rings, hence as a locally ringed space,

  2. as an object of the gros topos of sheaves on the site of commutative rings (with étale topology or Zariski topology) satisfying the condition that it is covered by representables via open maps.

In other words, a scheme may be identified with the sheaf it represents; this sheaf is often called the functor of points of the scheme.

To see this, note that by the Yoneda lemma a scheme may be identified with the sheaf it represents on the gros Zariski site of schemes; and since any scheme admits an affine open cover, the comparison lemma says that sheaves on the site of all schemes may be identified with sheaves on the site of affine schemes.

The functor of points approach has the advantage of making certain constructions much simpler (e.g. the fibered product in the category of schemes), and eliminating the need for certain constructions like the Zariski spectrum. In his famous 1973 Buffalo Colloquium talk, Alexander Grothendieck urged that his earlier definition of scheme via locally ringed spaces should be abandoned in favour of the functorial point of view.

Of course, the above discussion generalizes to other types of geometry and even higher geometry, the general perspective being known as synthetic differential geometry or similar. For discussion of functorial (higher) differential geometry see for instance at smooth set (smooth ∞-groupoid), for discussion of functorial supergeometry see at super formal smooth set (super formal smooth ∞-groupoid).

Example

The functor from commutative rings to sets which sends a ring, RR, to the set of simultaneous solutions in R nR^n of a set of polynomials, f 1,,f kf_1, \ldots, f_k in [t 1,,t n]\mathbb{Z}[t_1, \ldots,t_n] corresponds to the affine scheme X=Spec([t 1,,t n]/(f 1,,f k))X = Spec(\mathbb{Z}[t_1, \ldots,t_n]/(f_1, \ldots,f_k)). These RR-points are then equivalently the hom-space

Hom schemes(Spec(R),X). Hom_{schemes}(Spec(R), X).

The functor which sends RR to the points of the projective space R n\mathbb{P}^n_R corresponds to a non-affine scheme.

References

Revised on September 29, 2017 20:27:09 by Urs Schreiber (94.220.50.90)