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Found 76 declarations mentioning Algebra.Extension.CotangentSpace.

• Algebra.Extension.CotangentSpace 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) : Type (max w v)
• Algebra.Extension.cotangentComplex 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) : P.Cotangent →ₗ[S] P.CotangentSpace
• Algebra.Extension.toKaehler 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) : P.CotangentSpace →ₗ[S] Ω[S⁄R]
• Algebra.Extension.Hom.sub 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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') : P.CotangentSpace →ₗ[S] P'.Cotangent
• Algebra.Extension.H1Cotangent.val_zero 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} : ↑0 = 0
• Algebra.Generators.instFreeCotangentSpaceToExtension 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {ι : Type w} (P : Algebra.Generators R S ι) : Module.Free S P.toExtension.CotangentSpace
• Algebra.Generators.cotangentSpaceBasis 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {ι : Type w} (P : Algebra.Generators R S ι) : Module.Basis ι S P.toExtension.CotangentSpace
• Algebra.Extension.subsingleton_h1Cotangent 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) : Subsingleton P.H1Cotangent ↔ Function.Injective ⇑P.cotangentComplex
• Algebra.Extension.toKaehler_surjective 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} : Function.Surjective ⇑P.toKaehler
• Algebra.Extension.Cotangent.map_sub_map 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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.h1Cotangentι_ext 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} (x y : P.H1Cotangent) (e : ↑x = ↑y) : x = y
• Algebra.Extension.h1Cotangentι_ext_iff 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} {x y : P.H1Cotangent} : x = y ↔ ↑x = ↑y
• Algebra.Extension.h1Cotangentι_apply 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} (self : ↥P.cotangentComplex.ker) : Algebra.Extension.h1Cotangentι self = ↑self
• Algebra.Extension.exact_hCotangentι_cotangentComplex 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} : Function.Exact ⇑Algebra.Extension.h1Cotangentι ⇑P.cotangentComplex
• Algebra.Extension.CotangentSpace.map_id 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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.H1Cotangent.val_smul 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} {R₀ : Type u_1} [CommRing R₀] [Algebra R₀ S] [Module R₀ P.Cotangent] [IsScalarTower R₀ S P.Cotangent] (r : R₀) (x : P.H1Cotangent) : ↑(r • x) = r • ↑x
• Algebra.Extension.H1Cotangent.val_add 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} (x y : P.H1Cotangent) : ↑(x + y) = ↑x + ↑y
• Algebra.Extension.exact_cotangentComplex_toKaehler 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} : Function.Exact ⇑P.cotangentComplex ⇑P.toKaehler
• Algebra.Extension.H1Cotangent.map_apply_coe 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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') (c : ↥P.cotangentComplex.ker) : ↑((Algebra.Extension.H1Cotangent.map f) c) = (Algebra.Extension.Cotangent.map f) ↑c
• Algebra.Generators.cotangentSpaceBasis_apply 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {ι : Type w} (P : Algebra.Generators R S ι) (i : ι) : P.cotangentSpaceBasis i = 1 ⊗ₜ[P.Ring] (KaehlerDifferential.D R P.Ring) (MvPolynomial.X i)
• Algebra.Generators.toKaehler_cotangentSpaceBasis 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {ι : Type w} {P : Algebra.Generators R S ι} (i : ι) : P.toExtension.toKaehler (P.cotangentSpaceBasis i) = (KaehlerDifferential.D R S) (P.val i)
• Algebra.Generators.toKaehler_tmul_D 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {ι : Type w} {P : Algebra.Generators R S ι} (i : ι) : P.toExtension.toKaehler (1 ⊗ₜ[P.toExtension.Ring] (KaehlerDifferential.D R P.Ring) (MvPolynomial.X i)) = (KaehlerDifferential.D R S) (P.val i)
• Algebra.Extension.cotangentComplex_mk 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) (x : ↥P.ker) : P.cotangentComplex (Algebra.Extension.Cotangent.mk x) = 1 ⊗ₜ[P.Ring] (KaehlerDifferential.D R P.Ring) ↑x
• Algebra.Extension.Cotangent.mk_C_mem_ker_cotangentComplex 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {σ : Type u_1} (G : Algebra.Generators R S σ) {r : R} (hr : MvPolynomial.C r ∈ G.ker) : Algebra.Extension.Cotangent.mk ⟨MvPolynomial.C r, hr⟩ ∈ G.toExtension.cotangentComplex.ker
• Algebra.Extension.H1Cotangent.equiv_apply 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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 : ↥P₁.cotangentComplex.ker) : (Algebra.Extension.H1Cotangent.equiv f₁ f₂) c = ⟨(Algebra.Extension.Cotangent.map f₁) ↑c, ⋯⟩
• Algebra.Extension.CotangentSpace.map_cotangentComplex 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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.Hom.sub_one_tmul 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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') (x : P.Ring) : (f.sub g) (1 ⊗ₜ[P.Ring] (KaehlerDifferential.D R P.Ring) x) = Algebra.Extension.Cotangent.mk ((f.subToKer g) x)
• Algebra.Extension.CotangentSpace.map_tmul 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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.Hom.sub_tmul 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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') (r : S) (x : P.Ring) : (f.sub g) (r ⊗ₜ[P.Ring] (KaehlerDifferential.D R P.Ring) x) = r • Algebra.Extension.Cotangent.mk ((f.subToKer g) x)
• Algebra.Generators.cotangentSpaceBasis_repr_tmul 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {ι : Type w} (P : Algebra.Generators R S ι) (r : S) (x : P.Ring) (i : ι) : (P.cotangentSpaceBasis.repr (r ⊗ₜ[P.Ring] (KaehlerDifferential.D R P.Ring) x)) i = r * (MvPolynomial.aeval P.val) ((MvPolynomial.pderiv i) x)
• Algebra.Generators.cotangentSpaceBasis_repr_one_tmul 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {ι : Type w} (P : Algebra.Generators R S ι) (x : P.toExtension.Ring) (i : ι) : (P.cotangentSpaceBasis.repr (1 ⊗ₜ[P.toExtension.Ring] (KaehlerDifferential.D R P.toExtension.Ring) x)) i = (MvPolynomial.aeval P.val) ((MvPolynomial.pderiv i) x)
• Algebra.Extension.CotangentSpace.map_comp_cotangentComplex 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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_tmul_eq_tmul_map 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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 : Ω[P.Ring⁄R]) : (Algebra.Extension.CotangentSpace.map f) (x ⊗ₜ[P.Ring] y) = (algebraMap S S') x ⊗ₜ[P'.Ring] (KaehlerDifferential.map R R' P.Ring P'.Ring) y
• Algebra.Extension.CotangentSpace.map_comp_apply 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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.H1Cotangent.equiv_apply 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {ι : Type w} {ι' : Type u_1} (P : Algebra.Generators R S ι) (P' : Algebra.Generators R S ι') (c : ↥P.toExtension.cotangentComplex.ker) : (Algebra.Generators.H1Cotangent.equiv P P') c = ⟨(Algebra.Extension.Cotangent.map (P.defaultHom P').toExtensionHom) ↑c, ⋯⟩
• Algebra.Generators.repr_CotangentSpaceMap 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {ι : Type w} {P : Algebra.Generators R S ι} {R' : Type u'} {S' : Type v'} {ι' : Type w'} [CommRing R'] [CommRing S'] [Algebra R' S'] {P' : Algebra.Generators R' S' ι'} [Algebra R R'] [Algebra S S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] (f : P.Hom P') (i : ι) (j : ι') : (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.Extension.Cotangent.Basic
  {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_sub_map 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {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.Extension.cotangentComplexBaseChange_eq_lTensor_cotangentComplex 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) (A : Type u_1) [CommRing A] [Algebra S A] [Algebra P.Ring A] [IsScalarTower P.Ring S A] : KaehlerDifferential.cotangentComplexBaseChange R S P.Ring A = ↑(TensorProduct.AlgebraTensorModule.cancelBaseChange P.Ring S A A Ω[P.Ring⁄R]) ∘ₗ LinearMap.baseChange A P.cotangentComplex ∘ₗ ↑((TensorProduct.AlgebraTensorModule.cancelBaseChange P.Ring S A A ↥P.ker).symm ≪≫ₗ LinearEquiv.baseChange S A (TensorProduct P.Ring S ↥P.ker) P.Cotangent P.cotangentEquiv)
• Algebra.Extension.lTensor_cotangentComplex_eq_cotangentComplexBaseChange 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) (A : Type u_1) [CommRing A] [Algebra S A] [Algebra P.Ring A] [IsScalarTower P.Ring S A] : LinearMap.baseChange A P.cotangentComplex = ↑(TensorProduct.AlgebraTensorModule.cancelBaseChange P.Ring S A A Ω[P.Ring⁄R]).symm ∘ₗ KaehlerDifferential.cotangentComplexBaseChange R S P.Ring A ∘ₗ ↑((TensorProduct.AlgebraTensorModule.cancelBaseChange P.Ring S A A ↥P.ker).symm ≪≫ₗ LinearEquiv.baseChange S A (TensorProduct P.Ring S ↥P.ker) P.Cotangent P.cotangentEquiv).symm
• Algebra.Presentation.differentials.comm₂₃ 📋 Mathlib.Algebra.Module.Presentation.Differentials
  {R : Type u} {S : Type v} {ι : Type w} {σ : Type t} [CommRing R] [CommRing S] [Algebra R S] (pres : Algebra.Presentation R S ι σ) : pres.toExtension.toKaehler ∘ₗ ↑pres.cotangentSpaceBasis.repr.symm = pres.differentialsSolution.π
• Algebra.Presentation.differentials.comm₂₃' 📋 Mathlib.Algebra.Module.Presentation.Differentials
  {R : Type u} {S : Type v} {ι : Type w} {σ : Type t} [CommRing R] [CommRing S] [Algebra R S] (pres : Algebra.Presentation R S ι σ) : pres.toExtension.toKaehler ∘ₗ ↑pres.cotangentSpaceBasis.repr.symm = Finsupp.linearCombination S fun g => (KaehlerDifferential.D R S) (pres.val g)
• Algebra.Presentation.differentials.comm₁₂ 📋 Mathlib.Algebra.Module.Presentation.Differentials
  {R : Type u} {S : Type v} {ι : Type w} {σ : Type t} [CommRing R] [CommRing S] [Algebra R S] (pres : Algebra.Presentation R S ι σ) : pres.toExtension.cotangentComplex ∘ₗ Algebra.Presentation.differentials.hom₁ pres = ↑pres.cotangentSpaceBasis.repr.symm ∘ₗ pres.differentialsRelations.map
• Algebra.Presentation.differentials.comm₁₂_single 📋 Mathlib.Algebra.Module.Presentation.Differentials
  {R : Type u} {S : Type v} {ι : Type w} {σ : Type t} [CommRing R] [CommRing S] [Algebra R S] (pres : Algebra.Presentation R S ι σ) (r : σ) : pres.toExtension.cotangentComplex ((Algebra.Presentation.differentials.hom₁ pres) fun₀ | r => 1) = pres.cotangentSpaceBasis.repr.symm (pres.differentialsRelations.relation r)
• Algebra.Extension.CotangentSpace.map_toInfinitesimal_bijective 📋 Mathlib.RingTheory.Smooth.Kaehler
  {R : Type u_1} {S : Type u_3} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) : Function.Bijective ⇑(Algebra.Extension.CotangentSpace.map P.toInfinitesimal)
• Algebra.Extension.cotangentComplex_injective_iff 📋 Mathlib.RingTheory.Smooth.Basic
  {R : Type u} {A : Type v} [CommRing R] [CommRing A] [Algebra R A] (P : Algebra.Extension R A) [Algebra.FormallySmooth R P.Ring] : Function.Injective ⇑P.cotangentComplex ↔ Subsingleton (Algebra.H1Cotangent R A)
• Algebra.Extension.formallySmooth_iff_split_injection 📋 Mathlib.RingTheory.Smooth.Basic
  {R : Type u} {A : Type v} [CommRing R] [CommRing A] [Algebra R A] (P : Algebra.Extension R A) [Algebra.FormallySmooth R P.Ring] : Algebra.FormallySmooth R A ↔ ∃ l, l ∘ₗ P.cotangentComplex = LinearMap.id
• Algebra.Generators.H1Cotangent.δ_C 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) {r : S} (hr : MvPolynomial.C r ∈ Q.ker) : (Algebra.Generators.H1Cotangent.δ Q P) ⟨Algebra.Extension.Cotangent.mk ⟨MvPolynomial.C r, hr⟩, ⋯⟩ = 1 ⊗ₜ[S] (KaehlerDifferential.D R S) r
• Algebra.Generators.CotangentSpace.map_ofComp_surjective 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) : Function.Surjective ⇑(Algebra.Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom)
• Algebra.Generators.CotangentSpace.compEquiv 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) : (Q.comp P).toExtension.CotangentSpace ≃ₗ[T] Q.toExtension.CotangentSpace × TensorProduct S T P.toExtension.CotangentSpace
• Algebra.Generators.H1Cotangent.δ_eq_δAux 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) (x : ↥Q.ker) (hx : Algebra.Extension.Cotangent.mk x ∈ Q.toExtension.cotangentComplex.ker) : (Algebra.Generators.H1Cotangent.δ Q P) ⟨Algebra.Extension.Cotangent.mk x, hx⟩ = (Algebra.Generators.H1Cotangent.δAux R Q) ↑x
• Algebra.Generators.H1Cotangent.δAux_toAlgHom 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {ι' : Type w₃} {Q : Algebra.Generators S T ι} {Q' : Algebra.Generators S T ι'} (f : Q.Hom Q') (x : Q.Ring) : (Algebra.Generators.H1Cotangent.δAux R Q') (f.toAlgHom x) = (Algebra.Generators.H1Cotangent.δAux R Q) x + (Finsupp.linearCombination T (⇑(Algebra.Generators.H1Cotangent.δAux R Q') ∘ f.val)) (Q.cotangentSpaceBasis.repr (1 ⊗ₜ[Q.Ring] (KaehlerDifferential.D S Q.Ring) x))
• Algebra.Generators.H1Cotangent.map_comp_cotangentComplex_baseChange 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) : 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 u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) : Function.Injective ⇑(LinearMap.liftBaseChange T (Algebra.Extension.CotangentSpace.map (Q.toComp P).toExtensionHom))
• Algebra.Generators.CotangentSpace.exact 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) : 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 u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) : 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 u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) (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 u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) (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.H1Cotangent.δAux_ofComp 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) (x : (Q.comp P).Ring) : (Algebra.Generators.H1Cotangent.δAux R Q) ((Q.ofComp P).toAlgHom x) = (LinearMap.baseChange T P.toExtension.toKaehler) ((Algebra.Generators.CotangentSpace.compEquiv Q P) (1 ⊗ₜ[(Q.comp P).Ring] (KaehlerDifferential.D R (Q.comp P).Ring) x)).2
• Algebra.Generators.CotangentSpace.compEquiv_symm_inr 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) : ↑(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 u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) (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.tensorCotangentSpaceOfFormallyEtale 📋 Mathlib.RingTheory.Etale.Kaehler
  {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {P : Algebra.Extension R S} {Q : Algebra.Extension R T} (f : P.Hom Q) (H : f.toRingHom.FormallyEtale) : TensorProduct S T P.CotangentSpace ≃ₗ[T] Q.CotangentSpace
• Algebra.FormallySmooth.iff_injective_lTensor_residueField 📋 Mathlib.RingTheory.Smooth.Local
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsLocalRing S] [Algebra R S] (P : Algebra.Extension R S) [Algebra.FormallySmooth R P.Ring] [Module.Free P.Ring Ω[P.Ring⁄R]] [Module.Finite P.Ring Ω[P.Ring⁄R]] (h' : P.ker.FG) : Algebra.FormallySmooth R S ↔ Function.Injective ⇑(LinearMap.lTensor (IsLocalRing.ResidueField S) P.cotangentComplex)
• Algebra.SubmersivePresentation.sectionCotangent 📋 Mathlib.RingTheory.Smooth.StandardSmoothCotangent
  {R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} [CommRing R] [CommRing S] [Algebra R S] [Finite σ] (P : Algebra.SubmersivePresentation R S ι σ) : P.toExtension.CotangentSpace →ₗ[S] P.toExtension.Cotangent
• Algebra.SubmersivePresentation.sectionCotangent_comp 📋 Mathlib.RingTheory.Smooth.StandardSmoothCotangent
  {R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} [CommRing R] [CommRing S] [Algebra R S] [Finite σ] (P : Algebra.SubmersivePresentation R S ι σ) : P.sectionCotangent ∘ₗ P.toExtension.cotangentComplex = LinearMap.id
• Algebra.SubmersivePresentation.cotangentComplex_injective 📋 Mathlib.RingTheory.Smooth.StandardSmoothCotangent
  {R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} [CommRing R] [CommRing S] [Algebra R S] [Finite σ] (P : Algebra.SubmersivePresentation R S ι σ) : Function.Injective ⇑P.toExtension.cotangentComplex
• Algebra.SubmersivePresentation.sectionCotangent_zero_of_notMem_range 📋 Mathlib.RingTheory.Smooth.StandardSmoothCotangent
  {R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} [CommRing R] [CommRing S] [Algebra R S] [Finite σ] (P : Algebra.SubmersivePresentation R S ι σ) (i : ι) (hi : i ∉ Set.range P.map) : P.sectionCotangent (P.cotangentSpaceBasis i) = 0
• Algebra.SubmersivePresentation.sectionCotangent_eq_iff 📋 Mathlib.RingTheory.Smooth.StandardSmoothCotangent
  {R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} [CommRing R] [CommRing S] [Algebra R S] [Finite σ] (P : Algebra.SubmersivePresentation R S ι σ) (x : P.toExtension.CotangentSpace) (y : P.toExtension.Cotangent) : P.sectionCotangent x = y ↔ ∀ (i : σ), (P.cotangentSpaceBasis.repr x) (P.map i) = P.cotangentComplexAux y i
• Algebra.Generators.disjoint_ker_toKaehler_of_linearIndependent 📋 Mathlib.RingTheory.Extension.Cotangent.Free
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {ι : Type u_3} {κ : Type u_5} (P : Algebra.Generators R S ι) {v : κ → ι} (h : LinearIndependent S fun k => (KaehlerDifferential.D R S) (P.val (v k))) : Disjoint P.toExtension.toKaehler.ker (Submodule.span S (Set.range fun x => P.cotangentSpaceBasis (v x)))
• Algebra.Generators.cotangentRestrict_bijective_of_isCompl 📋 Mathlib.RingTheory.Extension.Cotangent.Free
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {ι : Type u_3} {σ : Type u_4} {κ : Type u_5} (P : Algebra.Generators R S ι) {u : σ → ι} (hu : Function.Injective u) {v : κ → ι} (huv : IsCompl (Set.range v) (Set.range u)) (hm : Submodule.span S (Set.range fun i => (KaehlerDifferential.D R S) (P.val (v i))) = ⊤) (hk : Disjoint P.toExtension.toKaehler.ker (Submodule.span S (Set.range fun x => P.cotangentSpaceBasis (v x)))) [Subsingleton (Algebra.H1Cotangent R S)] : Function.Bijective ⇑(P.cotangentRestrict hu)
• Algebra.Extension.tensorCotangentSpace 📋 Mathlib.RingTheory.Extension.Cotangent.BaseChange
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) (T : Type u_4) [CommRing T] [Algebra R T] : TensorProduct R T P.CotangentSpace ≃ₗ[T] P.baseChange.CotangentSpace
• Algebra.Extension.tensorToH1Cotangent_tmul 📋 Mathlib.RingTheory.Extension.Cotangent.BaseChange
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) (T : Type u_3) [CommRing T] [Algebra R T] (t : T) (x : P.H1Cotangent) : ↑((P.tensorToH1Cotangent T) (t ⊗ₜ[R] x)) = t • (Algebra.Extension.Cotangent.map (Algebra.Extension.toBaseChange T)) ↑x
• Algebra.Extension.tensorCotangentSpace_tmul_tmul 📋 Mathlib.RingTheory.Extension.Cotangent.BaseChange
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) (T : Type u_3) [CommRing T] [Algebra R T] (t : T) (s : S) (x : Ω[P.Ring⁄R]) : (P.tensorCotangentSpace T) (t ⊗ₜ[R] (s ⊗ₜ[P.Ring] x)) = t ⊗ₜ[R] s ⊗ₜ[P.baseChange.Ring] (KaehlerDifferential.map R T P.Ring P.baseChange.Ring) x
• Algebra.Extension.tensorCotangentSpace_tmul 📋 Mathlib.RingTheory.Extension.Cotangent.BaseChange
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) (T : Type u_3) [CommRing T] [Algebra R T] (t : T) (x : P.CotangentSpace) : (P.tensorCotangentSpace T) (t ⊗ₜ[R] x) = t • (Algebra.Extension.CotangentSpace.map (Algebra.Extension.toBaseChange T)) x
• Algebra.Extension.cotangentComplex_comp_h1CotangentEquivCotangent 📋 Mathlib.RingTheory.Extension.ExtendScalars
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) : P.cotangentComplex ∘ₗ ↑P.h1CotangentEquivCotangent = Algebra.H1Cotangent.δ R P.Ring S

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. 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.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

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. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c