Theorems · Theorem · commutative algebra
IsLocalRing.local_hom_TFAE
∀ {R : Type u_1} {S : Type u_2} [inst : CommSemiring R] [inst_1 : IsLocalRing R] [inst_2 : CommSemiring S]
[inst_3 : IsLocalRing S] (f : R →+* S),
[IsLocalHom f, ⇑f '' ↑(IsLocalRing.maximalIdeal R) ⊆ ↑(IsLocalRing.maximalIdeal S),
Ideal.map f (IsLocalRing.maximalIdeal R) ≤ IsLocalRing.maximalIdeal S,
IsLocalRing.maximalIdeal R ≤ Ideal.comap f (IsLocalRing.maximalIdeal S),
Ideal.comap f (IsLocalRing.maximalIdeal S) = IsLocalRing.maximalIdeal R].TFAEA ring homomorphism between local rings is a local ring hom iff it reflects units, i.e. any preimage of a unit is still a unit.
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement and proof · cited by 10,189
- SetLike.coestatement and proof · cited by 8,199
- Set.imagestatement and proof · cited by 5,609
- Idealstatement · cited by 4,748
- Ideal.mapstatement and proof · cited by 692
- Ideal.comapstatement and proof · cited by 443
- le_of_eqproof · cited by 366
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealstatement and proof · cited by 297
Cited by10
Results whose statement or proof uses this declaration.
- IsLocalHom.of_surjectiveproof · cited by 4
- IsLocalRing.maximalIdeal_comapproof · cited by 3
- Module.FaithfullyFlat.of_flat_of_isLocalHomproof · cited by 3
- Ideal.image_subset_nonunits_valuationSubringproof · cited by 2
- AdicCompletion.algebraMap_isLocalHom_of_fgproof · cited by 1
- IsLocalRing.isLocalHom_iff_comap_closedPointproof · cited by 1
- LocalSubring.map_maximalIdeal_eq_top_of_isMaxproof · cited by 1
- LocalSubring.mem_of_isMax_of_isIntegralproof · cited by 1
- Algebra.FormallyUnramified.isField_quotient_map_maximalIdealproof · cited by 1
- LocalSubring.exists_le_valuationSubring_of_isIntegrallyClosedInproof · cited by 1