Theorems · Definition · commutative algebra

IsLocalization.AtPrime.primeSpectrumOrderIso

{R : Type u_1} →
  [inst : CommSemiring R] →
    (S : Type u_2) →
      [inst_1 : CommSemiring S] →
        [inst_2 : Algebra R S] →
          (I : Ideal R) →
            [hI : I.IsPrime] →
              [IsLocalization.AtPrime S I] → PrimeSpectrum S ≃o ↑(Set.Iic { asIdeal := I, isPrime := hI })

The prime spectrum of the localization of a commutative ring R at a prime ideal I are in order-preserving bijection with the interval $(-∞, I]$ in the prime spectrum of R.

Defined in

Mathlib.RingTheory.Localization.AtPrime.Basic

Cited by

4 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraIdeal.IsPrimeIsLocalization.AtPrime