Theorems · Theorem · algebraic geometry

PrimeSpectrum.isQuotientMap_of_specializingMap

∀ {R : Type u_1} {S : Type u_2} [inst : CommSemiring R] [inst_1 : CommSemiring S] {f : R →+* S},
  Function.Surjective (PrimeSpectrum.comap f) →
    SpecializingMap (PrimeSpectrum.comap f) → Topology.IsQuotientMap (PrimeSpectrum.comap f)

If f : Spec S → Spec R is specializing and surjective, the topology on Spec R is the quotient topology induced by f.

Defined in: Mathlib.RingTheory.Spectrum.Prime.Topology
Cited by: 0 results in Mathlib
Foundations: Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommSemiring CommSemiring

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Setproof · cited by 53,352
CommSemiringstatement and proof · cited by 10,911
RingHomstatement and proof · cited by 10,189
Set.preimageproof · cited by 4,946
IsClosedproof · cited by 1,639
PrimeSpectrumstatement and proof · cited by 625
PrimeSpectrum.comapstatement and proof · cited by 199
IsClosed.preimageproof · cited by 138
Topology.IsQuotientMapstatement · cited by 124
Set.image_preimage_eqproof · cited by 38
SpecializingMapstatement and proof · cited by 21
PrimeSpectrum.continuous_comapproof · cited by 19

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