Theorems · Theorem · algebraic geometry
PrimeSpectrum.isClosed_singleton_iff_isMaximal
∀ {R : Type u} [inst : CommSemiring R] (x : PrimeSpectrum R), IsClosed {x} ↔ x.asIdeal.IsMaximal- Cited by
- 8 results in Mathlib
- Foundations
- Depth 81 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.
Cites17
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.coeproof · cited by 8,199
- Idealproof · cited by 4,748
- IsClosedstatement · cited by 1,639
- PrimeSpectrumstatement and proof · cited by 625
- Ideal.IsMaximalstatement and proof · cited by 452
- PrimeSpectrum.asIdealstatement and proof · cited by 333
- PrimeSpectrum.zeroLocusproof · cited by 164
- Ideal.IsMaximal.isPrimeproof · cited by 53
- Ideal.exists_le_maximalproof · cited by 47
- PrimeSpectrum.extproof · cited by 43
Cited by8
Results whose statement or proof uses this declaration.
- PrimeSpectrum.discreteTopology_iff_finite_and_krullDimLE_zeroproof · cited by 3
- PrimeSpectrum.isJacobsonRing_iff_jacobsonSpaceproof · cited by 2
- PrimeSpectrum.t1Space_iff_isFieldproof · cited by 2
- PrimeSpectrum.isOpen_singleton_tfae_of_isNoetherian_of_isJacobsonRingproof · cited by 2
- IsLocalRing.isClosed_singleton_closedPointproof · cited by 1
- Ideal.Fiber.lift_residueField_surjectiveproof · cited by 1
- PrimeSpectrum.comap_singleton_isClosed_of_surjectiveproof · cited by 0
- AlgebraicGeometry.IsAffineOpen.primeIdealOf_isMaximal_of_isClosedproof · cited by 0