Theorems · Theorem · algebraic geometry
PrimeSpectrum.discreteTopology_iff_finite_and_krullDimLE_zero
∀ {R : Type u} [inst : CommSemiring R],
DiscreteTopology (PrimeSpectrum R) ↔ Finite (PrimeSpectrum R) ∧ Ring.KrullDimLE 0 RThe prime spectrum of a commutative semiring has discrete Zariski topology iff it is finite and the semiring has Krull dimension zero or is trivial.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 83 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.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Idealproof · cited by 4,748
- Finitestatement and proof · cited by 3,029
- Ideal.IsPrimeproof · cited by 827
- PrimeSpectrumstatement and proof · cited by 625
- DiscreteTopologystatement and proof · cited by 373
- Ring.KrullDimLEstatement and proof · cited by 79
- PrimeSpectrum.isClosed_singleton_iff_isMaximalproof · cited by 8
- finite_of_compact_of_discreteproof · cited by 6
- Ring.KrullDimLE.mk₀proof · cited by 4
- discreteTopology_iff_forall_isClosedproof · cited by 2
- DiscreteTopology.of_finite_of_isClosed_singletonproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Algebra.QuasiFinite.iff_finite_comap_preimage_singletonproof · cited by 4
- PrimeSpectrum.discreteTopology_of_toLocalization_surjectiveproof · cited by 1