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Theorems · Theorem · commutative algebra

IsLocalRing.ResidueField.map_comp

∀ {R : Type u_1} {S : Type u_2} {T : Type u_3} [inst : CommRing R] [inst_1 : IsLocalRing R] [inst_2 : CommRing S]
  [inst_3 : IsLocalRing S] [inst_4 : CommRing T] [inst_5 : IsLocalRing T] (f : T →+* R) (g : R →+* S)
  [inst_6 : IsLocalHom f] [inst_7 : IsLocalHom g],
  IsLocalRing.ResidueField.map (g.comp f) = (IsLocalRing.ResidueField.map g).comp (IsLocalRing.ResidueField.map f)

The composite of two IsLocalRing.ResidueField.maps is the IsLocalRing.ResidueField.map of the composite.

Defined in
Mathlib.RingTheory.LocalRing.ResidueField.Basic
Cited by
3 results in Mathlib
Foundations
Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CommRingIsLocalRingCommRingIsLocalRingCommRingIsLocalRingIsLocalHomIsLocalHom
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

AlgebraicGeometry.LocallyRingedSpace.residueFieldMap_comp · cited by 1LocallyRingedSpace.residu…IsLocalRing.ResidueField.mapEquiv_trans · cited by 0ResidueField.mapEquiv_tra…IsLocalRing.ResidueField.map_map · cited by 0ResidueField.map_mapCommRing · cited by 17173CommRingRingHom · cited by 10189RingHomRingHom.comp · cited by 899RingHom.compIsLocalRing · cited by 339IsLocalRingRingHom.ext · cited by 331RingHom.extIsLocalRing.ResidueField · cited by 156IsLocalRing.ResidueFieldIsLocalHom · cited by 100IsLocalHomIdeal.Quotient.ringHom_ext · cited by 17Quotient.ringHom_extIsLocalRing.ResidueField.map · cited by 16ResidueField.mapResidueField.map_compCITED BYCITES

Cites9

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

Cited by3

Results whose statement or proof uses this declaration.