Theorems · Theorem · algebraic geometry
PrimeSpectrum.isClosed_iff_zeroLocus_ideal
∀ {R : Type u} [inst : CommSemiring R] (Z : Set (PrimeSpectrum R)), IsClosed Z ↔ ∃ I, Z = PrimeSpectrum.zeroLocus ↑I- Cited by
- 4 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- SetLike.coestatement and proof · cited by 8,199
- Idealstatement and proof · cited by 4,748
- IsClosedstatement · cited by 1,639
- Ideal.spanproof · cited by 948
- PrimeSpectrumstatement and proof · cited by 625
- PrimeSpectrum.zeroLocusstatement and proof · cited by 164
- PrimeSpectrum.zeroLocus_spanproof · cited by 17
- PrimeSpectrum.isClosed_iff_zeroLocusproof · cited by 4
Cited by4
Results whose statement or proof uses this declaration.
- PrimeSpectrum.isClosed_image_of_stableUnderSpecializationproof · cited by 6
- PrimeSpectrum.exists_isClosed_singleton_of_isJacobsonRingproof · cited by 1
- PrimeSpectrum.isClosed_iff_zeroLocus_radical_idealproof · cited by 0
- AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffineproof · cited by 0