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Theorems · Theorem · algebraic geometry

IsLocalRing.isLocalHom_iff_comap_closedPoint

∀ {R : Type u} [inst : CommSemiring R] [inst_1 : IsLocalRing R] {S : Type v} [inst_2 : CommSemiring S]
  [inst_3 : IsLocalRing S] (f : R →+* S),
  IsLocalHom f ↔ PrimeSpectrum.comap f (IsLocalRing.closedPoint S) = IsLocalRing.closedPoint R
Defined in
Mathlib.RingTheory.Spectrum.Prime.Topology
Cited by
1 results in Mathlib
Foundations
Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CommSemiringIsLocalRingCommSemiringIsLocalRing
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Around this declaration

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

IsLocalRing.comap_closedPoint · cited by 4IsLocalRing.comap_closedP…DFunLike.coe · cited by 62936DFunLike.coeCommSemiring · cited by 10911CommSemiringRingHom · cited by 10189RingHomSetLike.coe · cited by 8199SetLike.coeSet.image · cited by 5609Set.imageIdeal.map · cited by 692Ideal.mapPrimeSpectrum · cited by 625PrimeSpectrumIdeal.comap · cited by 443Ideal.comapIsLocalRing · cited by 339IsLocalRingIsLocalRing.maximalIdeal · cited by 297IsLocalRing.maximalIdealPrimeSpectrum.comap · cited by 199PrimeSpectrum.comapList.TFAE.out · cited by 177TFAE.outIsLocalHom · cited by 100IsLocalHomIsLocalRing.closedPoint · cited by 60IsLocalRing.closedPointIsLocalRing.local_hom_TFAE · cited by 10IsLocalRing.local_hom_TFAEIsLocalRing.isLocalHom_iff_co…CITED BYCITES

Cites16

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Cited by1

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