nLab
Zariski site

Idea
Definition
Properties
Related concepts
References

Idea

There is a little site notion of Zariski topology, and a big site notion. As for the little site notion: the Zariski topology on the set of prime ideals of a commutative ring $R$ is the smallest topology that contains, as open sets, sets of the form ${p prime : a \notin p}$ where $a$ ranges over elements of $R$ .

As for the big site notion, the Zariski topology is a coverage on the opposite category CRing $^{op}$ of commutative rings. This article is mainly about the big site notion.

Definition

For $R$ a commutative ring, write $Spec R \in {CRing}^{op}$ for its incarnation in the opposite category.

Definition

A family of morphisms ${Spec A_{i} \to Spec R}$ in ${CRing}^{op}$ is a Zariski-covering precisely if

each ring $A_{i}$ is the localization

$A_{i} = R [r_{i}^{- 1}]$ A_i = R[r_i^{-1}]

of $R$ at a single element $r_{i} \in R$
$Spec A_{i} \to Spec R$ is the canonical inclusion, dual to the canonical ring homomorphism $R \to R [r_{i}^{- 1}]$ ;
There exists ${f_{i} \in R}$ such that

$\sum_{i} f_{i} r_{i} = 1 .$ \sum_i f_i r_i = 1 \,.

Remark

Geometrically, one may think of

$r_{i}$ as a function on the space $Spec R$ ;
$R [r_{i}^{- 1}]$ as the open subset of points in this space on which the function is not 0;
the covering condition as saying that the functions form a partition of unity on $Spec R$ .

Definition

Let ${CRing}_{fp} ↪ CRing$ be the full subcategory on finitely presented objects. This inherity the Zariski coverage.

The sheaf topos over this site is the big topos version of the Zariski topos.

Properties

Proposition

The Zariski coverage is subcanoncial.

Proposition

${CRing}_{fp}^{op}$ is the syntactic category of the cartesian theory of commutative rings;
${CRing}_{fp}^{op}$ equipped with the Zariski topology is the syntactic site of the geometric theory of local rings.

Hence

the big Zariski topos, def. 1, is the classifying topos for local rings.
a locally ringed topos is equivalently a topos $ℰ$ equipped with a geometric morphism into the big Zariski topos.

See classifying topos and locally ringed topos for more details on this.