Loogle!

Result

Found 17 declarations mentioning Algebra.Extension.Cotangent.map.

• Algebra.Extension.Cotangent.map 📋 Mathlib.RingTheory.Extension
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} {R' : Type u_1} {S' : Type u_2} [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') : P.Cotangent →ₗ[S] P'.Cotangent
• Algebra.Extension.Cotangent.map_id 📋 Mathlib.RingTheory.Extension
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} : Algebra.Extension.Cotangent.map (Algebra.Extension.Hom.id P) = LinearMap.id
• Algebra.Extension.Cotangent.map_comp 📋 Mathlib.RingTheory.Extension
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} {R' : Type u_1} {S' : Type u_2} [CommRing R'] [CommRing S'] [Algebra R' S'] {P' : Algebra.Extension R' S'} {R'' : Type u_4} {S'' : Type u_5} [CommRing R''] [CommRing S''] [Algebra R'' S''] (P'' : Algebra.Extension R'' S'') [Algebra R R'] [Algebra R' R''] [Algebra R' S''] [Algebra S S'] [Algebra S' S''] [Algebra S S''] [Algebra R S'] [IsScalarTower R R' S'] [Algebra R R''] [IsScalarTower R R' R''] [IsScalarTower R' R'' S''] [Algebra R S''] [IsScalarTower R R'' S''] [IsScalarTower S S' S''] (f : P.Hom P') (g : P'.Hom P'') : Algebra.Extension.Cotangent.map (g.comp f) = ↑S (Algebra.Extension.Cotangent.map g) ∘ₗ Algebra.Extension.Cotangent.map f
• Algebra.Extension.Cotangent.map_mk 📋 Mathlib.RingTheory.Extension
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} {R' : Type u_1} {S' : Type u_2} [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') (x : ↥P.ker) : (Algebra.Extension.Cotangent.map f) (Algebra.Extension.Cotangent.mk x) = Algebra.Extension.Cotangent.mk ⟨f.toAlgHom ↑x, ⋯⟩
• Algebra.Extension.Cotangent.map_sub_map 📋 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'] [IsScalarTower R S S'] (f g : P.Hom P') : Algebra.Extension.Cotangent.map f - Algebra.Extension.Cotangent.map g = f.sub g ∘ₗ P.cotangentComplex
• Algebra.Extension.CotangentSpace.map_cotangentComplex 📋 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') (x : P.Cotangent) : (Algebra.Extension.CotangentSpace.map f) (P.cotangentComplex x) = P'.cotangentComplex ((Algebra.Extension.Cotangent.map f) x)
• Algebra.Extension.CotangentSpace.map_comp_cotangentComplex 📋 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 ∘ₗ P.cotangentComplex = ↑S P'.cotangentComplex ∘ₗ Algebra.Extension.Cotangent.map f
• Algebra.Extension.H1Cotangent.map_apply_coe 📋 Mathlib.RingTheory.Kaehler.CotangentComplex
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra 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'] {P : Algebra.Extension R S} (f : P.Hom P') (c : ↥(LinearMap.ker P.cotangentComplex)) : ↑((Algebra.Extension.H1Cotangent.map f) c) = (Algebra.Extension.Cotangent.map f) ↑c
• Algebra.Extension.H1Cotangent.equiv_apply 📋 Mathlib.RingTheory.Kaehler.CotangentComplex
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P₁ : Algebra.Extension R S} {P₂ : Algebra.Extension R S} (f₁ : P₁.Hom P₂) (f₂ : P₂.Hom P₁) (c : ↥(LinearMap.ker P₁.cotangentComplex)) : (Algebra.Extension.H1Cotangent.equiv f₁ f₂) c = ⟨(Algebra.Extension.Cotangent.map f₁) ↑c, ⋯⟩
• Algebra.Generators.H1Cotangent.equiv_apply 📋 Mathlib.RingTheory.Kaehler.CotangentComplex
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Generators R S) (P' : Algebra.Generators R S) (c : ↥(LinearMap.ker P.toExtension.cotangentComplex)) : (Algebra.Generators.H1Cotangent.equiv P P') c = ⟨(Algebra.Extension.Cotangent.map (P.defaultHom P').toExtensionHom) ↑c, ⋯⟩
• Algebra.Generators.Cotangent.surjective_map_ofComp 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {T : Type uT} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (Q : Algebra.Generators S T) (P : Algebra.Generators R S) : Function.Surjective ⇑(Algebra.Extension.Cotangent.map (Q.ofComp P).toExtensionHom)
• Algebra.Generators.Cotangent.exact 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {T : Type uT} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (Q : Algebra.Generators S T) (P : Algebra.Generators R S) : Function.Exact ⇑(LinearMap.liftBaseChange T (Algebra.Extension.Cotangent.map (Q.toComp P).toExtensionHom)) ⇑(Algebra.Extension.Cotangent.map (Q.ofComp P).toExtensionHom)
• Algebra.Generators.H1Cotangent.δ.eq_1 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {T : Type uT} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (Q : Algebra.Generators S T) (P : Algebra.Generators R S) : Algebra.Generators.H1Cotangent.δ Q P = SnakeLemma.δ' (LinearMap.baseChange T P.toExtension.cotangentComplex) (Q.comp P).toExtension.cotangentComplex Q.toExtension.cotangentComplex (LinearMap.liftBaseChange T (Algebra.Extension.Cotangent.map (Q.toComp P).toExtensionHom)) (Algebra.Extension.Cotangent.map (Q.ofComp P).toExtensionHom) ⋯ (LinearMap.liftBaseChange T (Algebra.Extension.CotangentSpace.map (Q.toComp P).toExtensionHom)) (Algebra.Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom) ⋯ ⋯ ⋯ Algebra.Extension.h1Cotangentι ⋯ (LinearMap.baseChange T P.toExtension.toKaehler) ⋯ ⋯ ⋯
• Algebra.Generators.H1Cotangent.map_comp_cotangentComplex_baseChange 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {T : Type uT} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (Q : Algebra.Generators S T) (P : Algebra.Generators R S) : LinearMap.liftBaseChange T (Algebra.Extension.CotangentSpace.map (Q.toComp P).toExtensionHom) ∘ₗ LinearMap.baseChange T P.toExtension.cotangentComplex = (Q.comp P).toExtension.cotangentComplex ∘ₗ LinearMap.liftBaseChange T (Algebra.Extension.Cotangent.map (Q.toComp P).toExtensionHom)
• Algebra.Generators.H1Cotangent.δ_eq 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {T : Type uT} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (Q : Algebra.Generators S T) (P : Algebra.Generators R S) (x : Q.toExtension.H1Cotangent) (y : (Q.comp P).toExtension.Cotangent) (hy : (Algebra.Extension.Cotangent.map (Q.ofComp P).toExtensionHom) y = ↑x) (z : TensorProduct S T P.toExtension.CotangentSpace) (hz : (LinearMap.liftBaseChange T (Algebra.Extension.CotangentSpace.map (Q.toComp P).toExtensionHom)) z = (Q.comp P).toExtension.cotangentComplex y) : (Algebra.Generators.H1Cotangent.δ Q P) x = (LinearMap.baseChange T P.toExtension.toKaehler) z
• Algebra.Extension.Cotangent.map_toInfinitesimal_bijective 📋 Mathlib.RingTheory.Smooth.Kaehler
  {R S : Type u} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) : Function.Bijective ⇑(Algebra.Extension.Cotangent.map P.toInfinitesimal)
• Algebra.Generators.liftBaseChange_injective_of_isLocalizationAway 📋 Mathlib.RingTheory.CotangentLocalizationAway
  {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [CommRing S] [Algebra R S] [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (g : S) [IsLocalization.Away g T] (P : Algebra.Generators R S) : Function.Injective ⇑(LinearMap.liftBaseChange T (Algebra.Extension.Cotangent.map ((Algebra.Generators.localizationAway g).toComp P).toExtensionHom))

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 bce1d65