# Contents

## 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 $\{p\; \text{prime}: a \notin p\}$ 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 spectrum of a commutative ring, hence equivalently for its incarnation in the opposite category.

###### Definition

For $S \subset R$ a multiplicative subset, write $R[S^{-1}]$ for the corresponding localization and

$Spec(R[S^{-1}]) \longrightarrow Spec(R)$

for the dual of the canonical ring homomorphism $R \to R[S^{-1}]$.

###### Remark

The maps as in def. 1 are open immersion, called the standard opens of $Spec(R)$.

(e.g. Stack Project, lemma 10.9.17).

###### 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

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

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

$\sum_i f_i r_i = 1 \,.$
###### 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 $CRing_{fp} \hookrightarrow CRing$ 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

Hence

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

fpqc-site $\to$ fppf-site $\to$ syntomic site $\to$ étale site $\to$ Nisnevich site $\to$ Zariski site

## References

Examples A2.1.11(f) and D3.1.11 in

Section VIII.6 of

Revised on November 24, 2013 20:20:44 by Urs Schreiber (89.204.135.99)