Contents

# Contents

## Idea

There is a little site notion of the 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. are not open immersions for arbitrary multiplicative subsets $S$ (see a MathOverflow discussion). They are for subsets of the form $S = \{ f^0, f^1, f^2, \ldots \}$, in which case they are called the standard opens of $Spec(R)$.

###### 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 inherits the Zariski coverage.

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

## Properties

### Points

The maximal ideal in $R$ correspond precisely to the closed points of the prime spectrum $Spec(R)$ in the Zariski topology.

### As a site

###### Proposition

The Zariski coverage is subcanonical.

###### Proposition

Hence

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

### Sheafification

If $F$ is a presheaf on $CRing^{op}$ and $F^{++}$ denotes its sheafification, then the canonical morphism $F(R) \to F^{++}(R)$ is an isomorphism for all local rings $R$. This follows from the explicit description of the plus construction and the fact that a local ring admits only the trivial covering.

## Kripke–Joyal semantics

Writing $R \models \varphi$ for the interpretation of a formula $\varphi$ of the internal language of the big Zariski topos over $Spec(R)$ with the Kripke–Joyal semantics, the forcing relation can be expressed as follows.

$\begin{array}{lcl} R \models x = y : F &\Longleftrightarrow& x = y \in F(R). \\ R \models \top &\Longleftrightarrow& 1 = 1 \in R. \\ R \models \bot &\Longleftrightarrow& 1 = 0 \in R. \\ R \models \phi \wedge \psi &\Longleftrightarrow& R \models \phi \,\text{and}\, R \models \psi. \\ R \models \phi \vee \psi &\Longleftrightarrow& \text{there is a partition}\, \sum_i f_i = 1 \in R \,\text{such that for all}\, i, R[f_i^{-1}] \models \phi \,or\, R[f_i^{-1}] \models \psi. \\ R \models \phi \Rightarrow \psi &\Longleftrightarrow& \text{for any}\, R\text{-algebra}\, S \,\text{it holds that}\, (S \models \phi) \,implies\, (S \models \psi). \\ R \models \forall x:F. \phi &\Longleftrightarrow& \text{for any}\, R\text{-algebra}\, S \,\text{and any element}\, x \in F(S) \,\text{it holds that}\, S \models \phi[x]. \\ R \models \exists x.F. \phi &\Longleftrightarrow& \text{there is a partition}\, \sum_i f_i = 1 \in R \,\text{such that for all}\, i, \text{there exists an element}\, x_i \in F(R[f_i^{-1}]) \,\text{such that}\, R[f_i^{-1}] \models \phi[x_i]. \end{array}$

The only difference to the Kripke–Joyal semantics of the little Zariski topos is that in the clauses for $\Rightarrow$ and $\forall$, one has to restrict to $R$-algebras $S$ of the form $S = R[f^{-1}]$.

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

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

Section VIII.6 of