nLab Zariski site

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

• $CRing_{fp}^{op}$ 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. , is the classifying topos for local rings.

• a locally ringed topos is equivalently a topos $\mathcal{E}$ equipped with a geometric morphism into the big Zariski topos.

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 $S = R[f^{-1}]$.

Related concepts

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

• spectrum of a commutative ring

References

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

• Peter Johnstone, Sketches of an Elephant

Section VIII.6 of

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

• The Stacks Project, chapter 33 Topologies on Schemes

• Nick Duncan, Gros and Petit Toposes, talk notes, 88th Peripatetic Seminar on Sheaves and Logic, pdf.

• Daniel Murfet, The Zariski Site

category: algebraic geometry