Theorems · Theorem · commutative algebra

KaehlerDifferential.ker_map

∀ (R : Type u) (S : Type v) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (A : Type u_2)
  (B : Type u_3) [inst_3 : CommRing A] [inst_4 : CommRing B] [inst_5 : Algebra R A] [inst_6 : Algebra A B]
  [inst_7 : Algebra S B] [inst_8 : Algebra R B] [inst_9 : IsScalarTower R A B] [inst_10 : IsScalarTower R S B]
  [inst_11 : SMulCommClass S A B],
  (KaehlerDifferential.map R S A B).ker =
    Submodule.map (Finsupp.linearCombination A ⇑(KaehlerDifferential.D R A))
      (Submodule.comap (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B))
        (Submodule.restrictScalars A (KaehlerDifferential.kerTotal S B)))

Defined in

Mathlib.RingTheory.Kaehler.Basic

Cited by

2 results in Mathlib

Foundations

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

Assumes

CommRingCommRingAlgebraCommRingCommRingAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerSMulCommClass