Loogle!

Result

Found 21 declarations mentioning Algebra.Extension.CotangentSpace.map.

• Algebra.Extension.CotangentSpace.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'] (f : P.Hom P') : P.CotangentSpace →ₗ[S] P'.CotangentSpace
• Algebra.Extension.CotangentSpace.map_id 📋 Mathlib.RingTheory.Kaehler.CotangentComplex
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} : Algebra.Extension.CotangentSpace.map (Algebra.Extension.Hom.id P) = LinearMap.id
• Algebra.Extension.CotangentSpace.map_tmul 📋 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 : S) (y : P.Ring) : (Algebra.Extension.CotangentSpace.map f) (x ⊗ₜ[P.Ring] (KaehlerDifferential.D R P.Ring) y) = (algebraMap S S') x ⊗ₜ[P'.Ring] (KaehlerDifferential.D R' P'.Ring) (f.toAlgHom y)
• 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.CotangentSpace.map_comp_apply 📋 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'] {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''] [Algebra R' R''] [Algebra S' S''] [Algebra R' S''] [IsScalarTower R' R'' S''] [IsScalarTower R R' R''] [IsScalarTower S S' S''] (f : P.Hom P') (g : P'.Hom P'') (x : P.CotangentSpace) : (Algebra.Extension.CotangentSpace.map (g.comp f)) x = (Algebra.Extension.CotangentSpace.map g) ((Algebra.Extension.CotangentSpace.map f) x)
• Algebra.Generators.repr_CotangentSpaceMap 📋 Mathlib.RingTheory.Kaehler.CotangentComplex
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Generators R S} {R' : Type u'} {S' : Type v'} [CommRing R'] [CommRing S'] [Algebra R' S'] {P' : Algebra.Generators R' S'} [Algebra R R'] [Algebra S S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] (f : P.Hom P') (i : P.vars) (j : P'.vars) : (P'.cotangentSpaceBasis.repr ((Algebra.Extension.CotangentSpace.map f.toExtensionHom) (P.cotangentSpaceBasis i))) j = (MvPolynomial.aeval P'.val) ((MvPolynomial.pderiv j) (f.val i))
• Algebra.Extension.CotangentSpace.map_comp 📋 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'] {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''] [Algebra R' R''] [Algebra S' S''] [Algebra R' S''] [IsScalarTower R' R'' S''] [IsScalarTower R R' R''] [IsScalarTower S S' S''] (f : P.Hom P') (g : P'.Hom P'') : Algebra.Extension.CotangentSpace.map (g.comp f) = ↑S (Algebra.Extension.CotangentSpace.map g) ∘ₗ Algebra.Extension.CotangentSpace.map f
• 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)
• Algebra.Extension.CotangentSpace.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.CotangentSpace.map f - Algebra.Extension.CotangentSpace.map g = ↑S P'.cotangentComplex ∘ₗ f.sub g
• Algebra.Generators.CotangentSpace.map_ofComp_surjective 📋 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.CotangentSpace.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.CotangentSpace.map_toComp_injective 📋 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.Injective ⇑(LinearMap.liftBaseChange T (Algebra.Extension.CotangentSpace.map (Q.toComp P).toExtensionHom))
• Algebra.Generators.CotangentSpace.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.CotangentSpace.map (Q.toComp P).toExtensionHom)) ⇑(Algebra.Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom)
• Algebra.Generators.CotangentSpace.fst_compEquiv 📋 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.fst T Q.toExtension.CotangentSpace (TensorProduct S T P.toExtension.CotangentSpace) ∘ₗ ↑(Algebra.Generators.CotangentSpace.compEquiv Q P) = Algebra.Extension.CotangentSpace.map (Q.ofComp 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.Generators.CotangentSpace.fst_compEquiv_apply 📋 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.comp P).toExtension.CotangentSpace) : ((Algebra.Generators.CotangentSpace.compEquiv Q P) x).1 = (Algebra.Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom) x
• Algebra.Generators.CotangentSpace.compEquiv_symm_inr 📋 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.CotangentSpace.compEquiv Q P).symm ∘ₗ LinearMap.inr T Q.toExtension.CotangentSpace (TensorProduct S T P.toExtension.CotangentSpace) = LinearMap.liftBaseChange T (Algebra.Extension.CotangentSpace.map (Q.toComp P).toExtensionHom)
• Algebra.Generators.CotangentSpace.compEquiv_symm_zero 📋 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 : TensorProduct S T P.toExtension.CotangentSpace) : (Algebra.Generators.CotangentSpace.compEquiv Q P).symm (0, x) = (LinearMap.liftBaseChange T (Algebra.Extension.CotangentSpace.map (Q.toComp P).toExtensionHom)) x
• Algebra.Extension.CotangentSpace.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.CotangentSpace.map P.toInfinitesimal)

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