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 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 .
The maps as in def. are not open immersions for arbitrary multiplicative subsets (see a MathOverflow discussion). They are for subsets of the form , in which case they are called the standard opens of .
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
Geometrically, one may think of
as a function on the space ;
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 .
Let 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.
The maximal ideal in correspond precisely to the closed points of the prime spectrum in the Zariski topology.
The Zariski coverage is subcanonical.
is the syntactic category of the cartesian theory of commutative rings;
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 equipped with a geometric morphism into the big Zariski topos.
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
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
Last revised on September 26, 2024 at 15:28:27. See the history of this page for a list of all contributions to it.