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 is the smallest topology that contains, as open sets, sets of the form where ranges over elements of .
As for the big site notion, the Zariski topology is a coverage on the opposite category CRing of commutative rings. This article is mainly about the big site notion.
For a commutative ring, write for its spectrum of a commutative ring, hence equivalently for its incarnation in the opposite category.
For a multiplicative subset, write for the corresponding localization and
for the dual of the canonical ring homomorphism .
A family of morphisms in is a Zariski-covering precisely if
each ring is the localization
of at a single element
is the canonical inclusion, dual to the canonical ring homomorphism ;
There exists such that
The maximal ideal in correspond precisely to the closed points of the prime spectrum in the Zariski topology.
As a site
See classifying topos and locally ringed topos for more details on this.
If is a presheaf on and denotes its sheafification, then the canonical morphism is an isomorphism for all local rings . This follows from the explicit description of the plus construction and the fact that a local ring admits only the trivial covering.
Writing for the interpretation of a formula of the internal language of the big Zariski topos over with the Kripke–Joyal semantics, the forcing relation can be expressed as follows.
The only difference to the Kripke–Joyal semantics of the little Zariski topos is that in the clauses for and , one has to restrict to -algebras of the form .
fpqc-site fppf-site syntomic site étale site Nisnevich site Zariski site
Examples A2.1.11(f) and D3.1.11 in
Section VIII.6 of