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Zariski site

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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 RR is the smallest topology that contains, as open sets, sets of the form {pprime:ap}\{p\; \text{prime}: a \notin p\} where aa ranges over elements of RR.

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

Definition

For RR a commutative ring, write SpecRCRing opSpec R \in CRing^{op} for its spectrum of a commutative ring, hence equivalently for its incarnation in the opposite category.

Definition

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

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

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

Remark

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

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

Definition

A family of morphisms {SpecA iSpecR}\{Spec A_i \to Spec R\} in CRing opCRing^{op} is a Zariski-covering precisely if

  • each ring A iA_i is the localization

    A i=R[r i 1] A_i = R[r_i^{-1}]

    of RR at a single element r iRr_i \in R

  • SpecA iSpecRSpec A_i \to Spec R is the canonical inclusion, dual to the canonical ring homomorphism RR[r i 1]R \to R[r_i^{-1}];

  • There exists {f iR}\{f_i \in R\} such that

    if ir i=1. \sum_i f_i r_i = 1 \,.
Remark

Geometrically, one may think of

  • r ir_i as a function on the space SpecRSpec R;

  • R[r i 1]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 SpecRSpec R.

Definition

Let CRing fpCRingCRing_{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 RR correspond precisely to the closed points of the prime spectrum Spec(R)Spec(R) in the Zariski topology.

As a site

Proposition

The Zariski coverage is subcanoncial.

Proposition

Hence

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

Kripke–Joyal semantics

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

Rx=y:F x=yF(R). R 1=1R. R 1=0R. Rϕψ RϕandRψ. Rϕψ there is a partition if i=1Rsuch that for alli,R[f i 1]ϕorR[f i 1]ψ. Rϕψ for anyR-algebraSit holds that(Sϕ)implies(Sψ). Rx:F.ϕ for anyR-algebraSand any elementxF(S)it holds thatSϕ[x]. Rx.F.ϕ there is a partition if i=1Rsuch that for alli,there exists an elementx iF(R[f i 1])such thatR[f i 1]ϕ[x i]. \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 RR-algebras SS of the form S=R[f 1]S = R[f^{-1}].

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 20, 2014 17:35:08 by Ingo Blechschmidt (137.250.162.16)