Theorems · Theorem · algebraic geometry
PrimeSpectrum.isCompact_isOpen_iff_ideal
∀ {R : Type u} [inst : CommSemiring R] {s : Set (PrimeSpectrum R)},
IsCompact s ∧ IsOpen s ↔ ∃ I, I.FG ∧ (PrimeSpectrum.zeroLocus ↑I)ᶜ = s- Cited by
- 1 results in Mathlib
- Foundations
- Depth 87 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.
Cites14
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
- Finsetproof · cited by 13,712
- CommSemiringstatement and proof · cited by 10,911
- SetLike.coestatement and proof · cited by 8,199
- Idealstatement and proof · cited by 4,748
- Compl.complstatement and proof · cited by 2,925
- IsOpenstatement · cited by 2,400
- IsCompactstatement · cited by 1,282
- Ideal.spanproof · cited by 948
- PrimeSpectrumstatement and proof · cited by 625
- PrimeSpectrum.zeroLocusstatement and proof · cited by 164
- Ideal.FGstatement and proof · cited by 99
Cited by1
Results whose statement or proof uses this declaration.
- PrimeSpectrum.exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopenproof · cited by 2