Theorems · Theorem · algebraic geometry
PrimeSpectrum.localization_away_comap_range
∀ {R : Type u} [inst : CommSemiring R] (S : Type v) [inst_1 : CommSemiring S] [inst_2 : Algebra R S] (r : R)
[IsLocalization.Away r S], Set.range (PrimeSpectrum.comap (algebraMap R S)) = ↑(PrimeSpectrum.basicOpen r)- Cited by
- 17 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
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
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- SetLike.coestatement and proof · cited by 8,199
- Set.ofPredproof · cited by 6,101
- Bot.botproof · cited by 4,720
- Algebra.algebraMapstatement · cited by 4,706
- Set.rangestatement · cited by 4,705
- Set.extproof · cited by 2,266
- TopologicalSpace.Opensstatement · cited by 2,040
- PrimeSpectrumstatement and proof · cited by 625
- Submonoid.powersproof · cited by 408
Cited by17
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.IsAffineOpen.basicOpenproof · cited by 19
- Algebra.IsLocalIso.of_span_range_eq_topproof · cited by 3
- PrimeSpectrum.isCompact_basicOpenproof · cited by 3
- AlgebraicGeometry.Scheme.exists_affine_mem_range_and_range_subsetproof · cited by 2
- Module.basicOpen_subset_freeLocus_iffproof · cited by 2
- Algebra.QuasiFiniteAt.exists_basicOpen_eq_singletonproof · cited by 2
- Algebra.isOpen_smoothLocusproof · cited by 2
- AlgebraicGeometry.Proj.awayι_preimage_basicOpenproof · cited by 1
- AlgebraicGeometry.exists_etale_isCompl_of_quasiFiniteAtproof · cited by 1
- AlgebraicGeometry.Scheme.IdealSheafData.opensRange_glueDataObjMapproof · cited by 1
- PrimeSpectrum.localization_away_isOpenEmbeddingproof · cited by 1