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The vanishing sets define the Zariski topology on the prime spectrum
Statement
Let be a commutative ring. The subsets , as ranges over the ideals of , contain and , are closed under arbitrary intersections and finite unions, and therefore define a topology on .
Facts & Assumptions
Given: A commutative ring .
The vanishing sets satisfy , , arbitrary intersections, and finite unions (Vanishing-set identities).
A family of subsets of a set that contains the whole set and the empty set, is closed under arbitrary intersections, and is closed under finite unions is the family of closed sets of a topology.
Proof
Fact [L1] gives exactly the four closed-set properties listed in the statement for the family of subsets of .
By [A1], any family with those four properties is the family of closed sets of a topology on the underlying set. Therefore the subsets define a topology on .
The vanishing sets are precisely the closed sets of the Zariski topology on .
Depends on
Used by
- The prime-spectrum construction is a contravariant functor to topological spaces Corollary
- The quasi-finite locus of a finite-type algebra is open Corollary
- A nonclosed open immersion is not proper Counterexample
- The affine line is not proper Counterexample
- A closed point immersion is unramified Example
- A finite algebra is its own Zariski Main factor Example
- Finite power map of the affine line Example
- Incidence projection has closed determinantal image Example
- Closed gluing of two projective three-spaces is proper Lemma
- Closed immersions are affine quotients and survive base change Lemma
- Every Zariski-open subset is a union of distinguished opens Lemma
- Fpqc covers are universally submersive Lemma
- The spectrum of a finite product ring is the disjoint union of the factor spectra Lemma
- The spectrum of a localisation is the subspace of primes disjoint from the denominator set Lemma
- The spectrum of a quotient is a closed subspace Lemma
- A quasi-finite algebra factors openly through a finite algebra Theorem
- Closed immersions into affine schemes are quotient spectra Theorem
- Finite morphisms are integral and universally closed Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. S. Milne, A Primer of Commutative Algebra, v4.03, Proposition 14.1 (standard reference, not scraped)
- The Stacks Project, Definition 10.17.3 (standard reference, not scraped)