Zariski site



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.


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.


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}].


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).


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 \,.

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.


Let CRing fpCRingCRing_{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.



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


The Zariski coverage is subcanoncial.



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


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

Section VIII.6 of

Revised on May 23, 2014 06:45:53 by Urs Schreiber (