The prime spectrum and vanishing sets

Depends on

• Prime ideals and maximal ideals in a commutative ring
• The ideal generated by a subset and principal ideals

Used by

• Maps from proper integral schemes to the affine line have closed-point image Corollary
• A nonclosed open immersion is not proper Counterexample
• Finite type need not be locally free Counterexample
• Quasi-separatedness in pushforward cannot be omitted Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• Geometric genus of a singular curve Definition
• Principal distinguished subsets of the prime spectrum Definition
• The underlying space of an affine spectrum Definition
• A coherent closed-point skyscraper Example
• Finite power map of the affine line Example
• Fitting ideals of a diagonal two-by-two presentation Example
• Incidence projection has closed determinantal image Example
• Quotient module sheaf and its support Example
• The zero ring has empty prime spectrum Example
• A regular point lies on one irreducible component Lemma
• Closed gluing of two projective three-spaces is proper Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Constructible subsets stable under generalisation are open in an affine spectrum Lemma
• Every Zariski-closed subset has a unique radical defining ideal Lemma
• Fpqc covers are universally submersive Lemma
• Function field of an integral finite-type scheme Lemma
• Morphisms from complete connected schemes to affine schemes are constant Lemma
• Noetherian devissage for coherent proper pushforward Lemma
• Primes of a localization avoid the denominator set Lemma
• Primes of a quotient lie over the kernel Lemma
• Regular hyperplane step for coherent support induction Lemma
• Schematic closure and agreement on a dense open Lemma
• The closure of a prime is its vanishing set Lemma
• The spectrum map pulls back vanishing sets Lemma
• The spectrum map respects composition and identities Lemma
• The spectrum of a finite product ring is the disjoint union of the factor spectra Lemma
• The vanishing sets define the Zariski topology on the prime spectrum Lemma
• Valuation lifts detect universal closedness Lemma
• Vanishing sets of arbitrary sums Lemma
• Vanishing sets of finite products Lemma
• Vanishing sets reverse inclusions Lemma
• Vanishing-set identities Lemma
• Global functions on proper integral schemes form a finite extension of the base field Theorem
• Irreducible components of the spectrum correspond to minimal prime ideals Theorem
• Smooth proper curves, dominant morphisms and function fields 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, §14 The spectrum of a ring (standard reference, not scraped)
• A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §13 The Spectrum of a Ring (standard reference, not scraped)
• The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)