Theorems · Theorem · algebraic geometry
PrimeSpectrum.isClopen_iff_zeroLocus
∀ {R : Type u} [inst : CommRing R] {s : Set (PrimeSpectrum R)},
IsClopen s ↔ ∃ e, IsIdempotentElem e ∧ s = PrimeSpectrum.zeroLocus {e}- Cited by
- 1 results in Mathlib
- Foundations
- Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
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.
- Setstatement and proof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- SetLike.coeproof · cited by 8,199
- PrimeSpectrumstatement and proof · cited by 625
- IsIdempotentElemstatement and proof · cited by 217
- IsClopenstatement · cited by 189
- PrimeSpectrum.zeroLocusstatement and proof · cited by 164
- PrimeSpectrum.basicOpenproof · cited by 163
- IsIdempotentElem.one_subproof · cited by 20
- PrimeSpectrum.basicOpen_eq_zeroLocus_of_isIdempotentElemproof · cited by 3
- PrimeSpectrum.isClopen_iffproof · cited by 2
- PrimeSpectrum.zeroLocus_eq_basicOpen_of_isIdempotentElemproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.isOpenImmersion_SpecMap_iff_of_surjectiveproof · cited by 0