Theorems · Theorem · commutative algebra
IsLocalization.AtPrime.under_maximalIdeal
∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : CommSemiring S] [inst_2 : Algebra R S] (I : Ideal R)
[hI : I.IsPrime] [inst_3 : IsLocalization.AtPrime S I] (h : optParam (IsLocalRing S) ⋯),
Ideal.under R (IsLocalRing.maximalIdeal S) = I- Cited by
- 7 results in Mathlib
- Foundations
- Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- Ideal.IsPrimestatement and proof · cited by 827
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealstatement · cited by 297
- Ideal.understatement · cited by 170
- Ideal.extproof · cited by 131
- IsLocalization.AtPrimestatement and proof · cited by 79
- IsLocalization.AtPrime.isLocalRingstatement and proof · cited by 19
- IsLocalization.AtPrime.to_map_mem_maximal_iffproof · cited by 9
Cited by7
Results whose statement or proof uses this declaration.
- Localization.AtPrime.under_maximalIdealproof · cited by 9
- IsLocalization.AtPrime.map_eq_maximalIdealproof · cited by 8
- IsLocalization.AtPrime.ringKrullDim_eq_heightproof · cited by 4
- Algebra.QuasiFiniteAt.exists_basicOpen_eq_singletonproof · cited by 2
- IsLocalization.AtPrime.radical_map_of_mem_minimalPrimesproof · cited by 1
- IsLocalization.AtPrime.comap_maximalIdealproof · cited by 0