nLab schemes as sheaves on affine schemes

Idea

Grothendieck introduced spaces as locally representable sheaves over a fixed site whose objects are interpreted local models. If local models are affine schemes, then algebraic schemes (and, more generally, algebraic spaces) can be considered as an example.

Schemes as presheaves

Given a ring $R$, the correspondence $Y\mapsto Spec(\Gamma_Y \mathcal{O}_Y)$ extends to a functor $Scheme\to Aff$. A basic result, sometimes referred to as fundamental theorem on morphisms of schemes says that there is a bijection

$$CRing(R, \Gamma_Y\mathcal{O}_Y) \cong Scheme(Y, Spec R).$$

More precisely, for fixed $Y$, and for varying $R$ there is a restricted functor

$$Scheme(-,Y)|_{Aff^{op}} = h_Y|_{Aff^{op}} = h_Y|_{CRing} : CRing\to Set,$$

and the functor $Y\mapsto h_Y|_{CRing}$ from schemes to presheaves on $Aff = CRing^{op}$ is fully faithful. Thus the general schemes if defined as ringed spaces, indeed form a full subcategory of the category of presheaves on $Aff$.

In fact, the theorem is more general: instead of $Y$ being a scheme, we can take any locally ringed space. Thus the entire category of locally ringed spaces embeds into the category of presheaves on $Aff$.

Schemes as locally representable sheaves

In fact, one can characterize the essential image of the category of schemes in the category of presheaves over the category of affine schemes, as the full subcategory spanned by those presheaves which are $\tau$-locally representable and which are $\tau$-sheaves, where $\tau$ is the Zariski Grothendieck topology on the category of affine schemes.

See Demazure, Gabriel for functorial approach to schemes, and relations to standard spectra.

Noncommutative analogues

…(Rosenberg 1998 for fundamental theorem on morphisms, and 1999 for the equivalence of two approaches)

Derived analogues

…(Toen, Vezzosi etc.)

References

Related entries in $n$Lab

• affine scheme, algebraic scheme
• morphism of schemes
• Yoneda embedding, Isbell duality
• functorial geometry

Web sources

For the fundamental theorem of morphisms of schemes see for example Theorem 2 in MIT 18-726 AG course notes

• here

Literature

A standard reference to the functorial approach to schemes

• Michel Demazure, Pierre Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970; English edition is Introduction to algebraic geometry and algebraic groups, North-Holland, Amsterdam 1980 (North-Holland)

The fundamental theorem of morphisms of schemes has been adapted to noncommutative schemes in

• Alexander Rosenberg, Noncommutative schemes, Compos. Math. 112 (1998) 93–125 doi