Theorems · Theorem · commutative algebra

AdicCompletion.maximalIdeal_eq_map_of_fg

∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsLocalRing R] (fg : (IsLocalRing.maximalIdeal R).FG),
  IsLocalRing.maximalIdeal (AdicCompletion (IsLocalRing.maximalIdeal R) R) =
    Ideal.map (algebraMap R (AdicCompletion (IsLocalRing.maximalIdeal R) R)) (IsLocalRing.maximalIdeal R)

Defined in: Mathlib.RingTheory.AdicCompletion.LocalRing
Cited by: 3 results in Mathlib
Foundations: Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing IsLocalRing

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Cites13

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CommRingstatement and proof · cited by 17,173
RingHomstatement · cited by 10,189
Idealstatement · cited by 4,748
Algebra.algebraMapstatement · cited by 4,706
le_reflproof · cited by 2,061
Ideal.mapstatement · cited by 692
IsLocalRingstatement and proof · cited by 339
IsLocalRing.maximalIdealstatement and proof · cited by 297
AdicCompletionstatement · cited by 160
Ideal.FGstatement and proof · cited by 99
IsLocalRing.eq_maximalIdealproof · cited by 20
AdicCompletion.isLocalRing_of_fgstatement · cited by 4

Cited by3

Results whose statement or proof uses this declaration.

AdicCompletion.residueField_map_bijective_of_fgproof · cited by 1
AdicCompletion.algebraMap_isLocalHom_of_fgproof · cited by 1
AdicCompletion.isAdicComplete_of_fgproof · cited by 0