Theorems · Theorem · algebraic geometry
PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure
∀ {R : Type u} [inst : CommSemiring R] (t : Set (PrimeSpectrum R)),
PrimeSpectrum.zeroLocus ↑(PrimeSpectrum.vanishingIdeal t) = closure t- Cited by
- 11 results in Mathlib
- Foundations
- Depth 80 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.
Cites18
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
- closurestatement and proof · cited by 1,254
- PrimeSpectrumstatement and proof · cited by 625
- subset_closureproof · cited by 309
- isClosed_closureproof · cited by 195
- PrimeSpectrum.zeroLocusstatement and proof · cited by 164
- IsClosed.closure_subset_iffproof · cited by 59
- PrimeSpectrum.vanishingIdealstatement and proof · cited by 36
- GaloisConnection.u_l_u_eq_uproof · cited by 17
Cited by11
Results whose statement or proof uses this declaration.
- PrimeSpectrum.isClosed_singleton_iff_isMaximalproof · cited by 8
- PrimeSpectrum.vanishingIdeal_closureproof · cited by 3
- PrimeSpectrum.vanishingIdeal_anti_mono_iffproof · cited by 2
- PrimeSpectrum.closure_range_comapproof · cited by 2
- PrimeSpectrum.vanishingIdeal_mem_minimalPrimesproof · cited by 2
- PrimeSpectrum.le_iff_mem_closureproof · cited by 1
- PrimeSpectrum.zeroLocus_minimalPrimesproof · cited by 1
- PrimeSpectrum.isIrreducible_iff_vanishingIdeal_isPrimeproof · cited by 1
- AlgebraicGeometry.genericPoint_eq_bot_of_affineproof · cited by 1
- PrimeSpectrum.isRadical_vanishingIdealproof · cited by 1
- PrimeSpectrum.closure_singletonproof · cited by 0