Theorems · Inductive type · algebraic geometry
PrimeSpectrum
(R : Type u_1) → [CommSemiring R] → Type u_1
The prime spectrum of a commutative (semi)ring R is the type of all prime ideals of R.
It is naturally endowed with a topology (the Zariski topology),
and a sheaf of commutative rings (see Mathlib/AlgebraicGeometry/StructureSheaf.lean).
It is a fundamental building block in algebraic geometry.
- Defined in
- Mathlib.RingTheory.Spectrum.Prime.Defs
- Cited by
- 625 results in Mathlib
- Foundations
- Depth 1 from the axioms, rests on 2 definitions · uses no axioms
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites1
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement · cited by 10,911
Cited by715
Results whose statement or proof uses this declaration.
- PrimeSpectrum.asIdealstatement and proof · cited by 333
- PrimeSpectrum.comapstatement and proof · cited by 199
- PrimeSpectrum.zeroLocusstatement and proof · cited by 164
- PrimeSpectrum.basicOpenstatement and proof · cited by 163
- AlgebraicGeometry.PrimeSpectrum.Topproof · cited by 104
- Ring.KrullDimLEproof · cited by 79
- ringKrullDimproof · cited by 75
- IsLocalRing.closedPointstatement · cited by 60
- Module.supportstatement and proof · cited by 52
- PrimeSpectrum.extstatement and proof · cited by 43
- Module.rankAtStalkstatement and proof · cited by 41
- PrimeSpectrum.vanishingIdealstatement and proof · cited by 36
Showing the 200 most cited of 715.