Loogle!

Result

Found 19 declarations mentioning KaehlerDifferential.map.

• KaehlerDifferential.map 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra S B] [Algebra R B] [IsScalarTower R A B] [IsScalarTower R S B] [SMulCommClass S A B] : Ω[A⁄R] →ₗ[A] Ω[B⁄S]
• KaehlerDifferential.map_surjective 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (B : Type u_3) [CommRing B] [Algebra S B] [Algebra R B] [IsScalarTower R S B] : Function.Surjective ⇑(KaehlerDifferential.map R S B B)
• KaehlerDifferential.map_surjective_of_surjective 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra S B] [Algebra R B] [IsScalarTower R A B] [IsScalarTower R S B] [SMulCommClass S A B] (h : Function.Surjective ⇑(algebraMap A B)) : Function.Surjective ⇑(KaehlerDifferential.map R S A B)
• KaehlerDifferential.map_D 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra S B] [Algebra R B] [IsScalarTower R A B] [IsScalarTower R S B] [SMulCommClass S A B] (x : A) : (KaehlerDifferential.map R S A B) ((KaehlerDifferential.D R A) x) = (KaehlerDifferential.D S B) ((algebraMap A B) x)
• KaehlerDifferential.exact_mapBaseChange_map 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] : Function.Exact ⇑(KaehlerDifferential.mapBaseChange R A B) ⇑(KaehlerDifferential.map R A B B)
• KaehlerDifferential.mapBaseChange_tmul 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (x : B) (y : Ω[A⁄R]) : (KaehlerDifferential.mapBaseChange R A B) (x ⊗ₜ[A] y) = x • (KaehlerDifferential.map R R A B) y
• KaehlerDifferential.range_mapBaseChange 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] : LinearMap.range (KaehlerDifferential.mapBaseChange R A B) = LinearMap.ker (KaehlerDifferential.map R A B B)
• KaehlerDifferential.ker_map 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra S B] [Algebra R B] [IsScalarTower R A B] [IsScalarTower R S B] [SMulCommClass S A B] : LinearMap.ker (KaehlerDifferential.map R S A B) = 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)))
• KaehlerDifferential.ker_map_of_surjective 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (h : Function.Surjective ⇑(algebraMap A B)) : LinearMap.ker (KaehlerDifferential.map R R A B) = Submodule.map (Finsupp.linearCombination A ⇑(KaehlerDifferential.D R A)) (LinearMap.ker (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B)))
• KaehlerDifferential.map_compDer 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra S B] [Algebra R B] [IsScalarTower R A B] [IsScalarTower R S B] [SMulCommClass S A B] : (KaehlerDifferential.map R S A B).compDer (KaehlerDifferential.D R A) = Derivation.compAlgebraMap A (Derivation.restrictScalars R (KaehlerDifferential.D S B))
• Algebra.Extension.CotangentSpace.map.eq_1 📋 Mathlib.RingTheory.Kaehler.CotangentComplex
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} {R' : Type u'} {S' : Type v'} [CommRing R'] [CommRing S'] [Algebra R' S'] {P' : Algebra.Extension R' S'} [Algebra R R'] [Algebra S S'] [Algebra R S'] [IsScalarTower R R' S'] (f : P.Hom P') : Algebra.Extension.CotangentSpace.map f = LinearMap.liftBaseChange S (↑P.Ring ((TensorProduct.mk P'.Ring S' (Ω[P'.Ring⁄R'])) 1) ∘ₗ KaehlerDifferential.map R R' P.Ring P'.Ring)
• KaehlerDifferential.isLocalizedModule_map 📋 Mathlib.RingTheory.Etale.Kaehler
  (R S T : Type u) [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (M : Submonoid S) [IsLocalization M T] : IsLocalizedModule M (KaehlerDifferential.map R R S T)
• KaehlerDifferential.isBaseChange_of_formallyEtale 📋 Mathlib.RingTheory.Etale.Kaehler
  (R S T : Type u) [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T] [Algebra.FormallyEtale S T] : IsBaseChange T (KaehlerDifferential.map R R S T)
• KaehlerDifferential.isLocalizedModule 📋 Mathlib.RingTheory.Kaehler.TensorProduct
  (R : Type u_1) (S : Type u_2) (A : Type u_3) (B : Type u_4) [CommRing R] [CommRing S] [Algebra R S] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] (p : Submonoid R) [IsLocalization p S] [IsLocalization (Algebra.algebraMapSubmonoid A p) B] : IsLocalizedModule p (↑R (KaehlerDifferential.map R S A B))
• KaehlerDifferential.isLocalizedModule_of_isLocalizedModule 📋 Mathlib.RingTheory.Kaehler.TensorProduct
  (R : Type u_1) (S : Type u_2) (A : Type u_3) (B : Type u_4) [CommRing R] [CommRing S] [Algebra R S] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] (p : Submonoid R) [IsLocalization p S] [IsLocalizedModule p (IsScalarTower.toAlgHom R A B).toLinearMap] : IsLocalizedModule p (↑R (KaehlerDifferential.map R S A B))
• KaehlerDifferential.isBaseChange 📋 Mathlib.RingTheory.Kaehler.TensorProduct
  (R : Type u_1) (S : Type u_2) (A : Type u_3) (B : Type u_4) [CommRing R] [CommRing S] [Algebra R S] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] [h : Algebra.IsPushout R S A B] : IsBaseChange S (↑R (KaehlerDifferential.map R S A B))
• KaehlerDifferential.tensorKaehlerEquiv_tmul 📋 Mathlib.RingTheory.Kaehler.TensorProduct
  (R : Type u_1) (S : Type u_2) (A : Type u_3) (B : Type u_4) [CommRing R] [CommRing S] [Algebra R S] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] [Algebra.IsPushout R S A B] (a : S) (b : Ω[A⁄R]) : (KaehlerDifferential.tensorKaehlerEquiv R S A B) (a ⊗ₜ[R] b) = a • (KaehlerDifferential.map R S A B) b
• KaehlerDifferential.map_liftBaseChange_smul 📋 Mathlib.RingTheory.Kaehler.TensorProduct
  (R : Type u_1) (S : Type u_2) (A : Type u_3) (B : Type u_4) [CommRing R] [CommRing S] [Algebra R S] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] [h : Algebra.IsPushout R S A B] (b : B) (x : TensorProduct R S (Ω[A⁄R])) : (LinearMap.liftBaseChange S (↑R (KaehlerDifferential.map R S A B))) (b • x) = b • (LinearMap.liftBaseChange S (↑R (KaehlerDifferential.map R S A B))) x
• KaehlerDifferential.tensorKaehlerEquiv_left_inv 📋 Mathlib.RingTheory.Kaehler.TensorProduct
  (R : Type u_1) (S : Type u_2) (A : Type u_3) (B : Type u_4) [CommRing R] [CommRing S] [Algebra R S] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] [Algebra.IsPushout R S A B] : ↑S (KaehlerDifferential.derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (KaehlerDifferential.map R S A B)) = LinearMap.id

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision 40fea08