nLab schemes as sheaves on affine schemes

Context

Geometry

Idea
Schemes as small presheaves
Schemes as locally representable sheaves
Noncommutative analogues
Derived analogues
References

Related entries

Literature

Idea

Grothendieck 1965 understood spaces as locally representable sheaves over a fixed site whose objects are interpreted as local models (“functorial geometry”). If local models are affine schemes, then schemes and algebraic spaces can be considered as an example.

Schemes as small presheaves

Given a commutative ground ring $R$ , let $Scheme$ denote the category of schemes (over $R$ ) regarded as locally ringed topological spaces, and let $Aff \hookrightarrow Scheme$ denote its full subcategory of affine schemes.

The assignment which takes $Y \in Scheme$ to the spectrum $Spec(-)$ of the global sections $\Gamma_Y(-)$ of its structure sheaf $\mathcal{O}_Y$

Y \mapsto Spec\big( \Gamma_Y (\mathcal{O}_Y) \big)

extends to a functor

Scheme \longrightarrow Aff \,.

A basic result, sometimes referred to as the fundamental theorem on morphisms of schemes, says that there is a bijection

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

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

Scheme(-,Y)\vert_{Aff^{op}} \;=\; h_Y|_{Aff^{op}} \;=\; h_Y|_{CRing} \,\colon\, CRing \longrightarrow Set \,,

and the functor

Y \,\mapsto\, h_Y|_{CRing}

from schemes to small presheaves on $Aff = CRing^{op}$ is fully faithful. Thus, schemes in the sense of ringed spaces, indeed form a full subcategory of the category of small 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 small presheaves on $Aff$ .

Schemes as locally representable sheaves

One can characterize the essential image of the category of schemes in the category of small presheaves over the category of affine schemes, as the full subcategory spanned by those small presheaves that are $\tau$ -locally representable and are $\tau$ -sheaves, where $\tau$ is the Zariski topology on $Aff$ .

More precisely, let $AffSch \coloneqq CRing^{op}$ denote the category of affine schemes but now defined as the opposite category of that of commutative (unital) rings. Define the Zariski topology on $AffSch$ as the Grothendieck topology generated by the coverage given by families

\{Spec R[r^{-1}] \to Spec R\mid r\in R\}

for all $R\in AffSch$ .

Now consider the category of small sheaves on the site $AffSch$ , i.e., the full subcategory of the category of small presheaves on $AffSch$ that satisfy the gluing property with respect to every covering family in the coverage.

A small presheaf $F$ is locally representable if there is a set-indexed covering family

f_i\colon X_i\to F,

where $X_i$ is the representable sheaf of an affine scheme and the morphisms $f_i$ are open immersions.

Open immersions of small presheaves are defined as follows. A morphism of small presheaves

F\to G

is an open immersion if for every small presheaf $X$ represented by an affine scheme and every morphism

X\to G,

the base change

X\times_G F\to X

is an open immersion.

See Demazure & Gabriel 1970 for the functorial approach to schemes and its relation to locally ringed spaces.

Noncommutative analogues

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

Derived analogues

…(Toen, Vezzosi etc.)

References

Related entries

Literature

The original observation (cf. functorial geometry):

Alexander Grothendieck: Introduction au langage fonctoriel, course in Algiers in November 1965, lecture notes by Max Karoubi [pdf scan, pdf]
Alexander Grothendieck: Introduction to functorial algebraic geometry, part 1: affine algebraic geometry, summer school in Buffalo (1973) lecture notes by Federico Gaeta [pdf scan, pdf]

Review:

Michel Demazure, Pierre Gabriel, Groupes algebriques, tome 1, avec un appendice Corps de classes local par Hazewinkel M, Mason and Cie, Paris (1970)

English translation:

Michel Demazure, Peter Gabriel, Introduction to algebraic geometry and algebraic groups, North-Holland mathematics studies 39 (1980) [ISBN:9780080871509]