Theorems · Definition · commutative algebra

Algebra.Generators.compLocalizationAwayAlgHom

{R : Type u_1} →
  {S : Type u_2} →
    (T : Type u_3) →
      {ι : Type u_4} →
        [inst : CommRing R] →
          [inst_1 : CommRing S] →
            [inst_2 : Algebra R S] →
              [inst_3 : CommRing T] →
                [inst_4 : Algebra R T] →
                  [inst_5 : Algebra S T] →
                    [inst_6 : IsScalarTower R S T] →
                      (g : S) →
                        [inst_7 : IsLocalization.Away g T] →
                          (P : Algebra.Generators R S ι) →
                            ((Algebra.Generators.localizationAway T g).comp P).Ring →ₐ[R]
                              Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g))

If R[X] → S generates S, T is the localization of S away from g and f is a pre-image of g in R[X], this is the R-algebra map R[X,Y] →ₐ[R] (R[X]/I²)[1/f] defined via mapping Y to 1/f.

Defined in

Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Cited by

5 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingAlgebraAlgebraIsScalarTowerIsLocalization.Away