theorem closedImmersion_stalkMap_surjective {X Z : AlgebraicGeometry.Scheme} (i : Z ⟶ X) [AlgebraicGeometry.IsClosedImmersion i] (z : ↥Z) : Function.Surjective ⇑(CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.stalkMap i z))

For a closed immersion i : Z → X, the induced map on stalks 𝒪_{X,i(z)} → 𝒪_{Z,z} is surjective at every point.

theorem conormalSequence_exact (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A] (B : Type u) [CommRing B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] (hsurj : Function.Surjective ⇑(algebraMap A B)) : (KaehlerDifferential.kerCotangentToTensor R A B).range = Submodule.restrictScalars A (KaehlerDifferential.mapBaseChange R A B).ker

Proposition 33 / 35 (Lecture 20, conormal sequence — exactness at the middle term). For a surjection A → B of R-algebras, the image of I/I² → B ⊗_A Ω_{A/R} equals the kernel of B ⊗_A Ω_{A/R} → Ω_{B/R}.

theorem conormalSequence_surjective (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A] (B : Type u) [CommRing B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] : Function.Surjective ⇑(KaehlerDifferential.map R A B B)

The natural map Ω_{A/R} ⊗_A B → Ω_{B/R} (equivalently, surjectivity at the right of the conormal/Jacobi–Zariski sequence) is surjective.

theorem jacobiZariski_exact (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A] (B : Type u) [CommRing B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] : Function.Exact ⇑(KaehlerDifferential.mapBaseChange R A B) ⇑(KaehlerDifferential.map R A B B)

Exactness of the Jacobi–Zariski sequence in the middle: B ⊗_A Ω_{A/R} → Ω_{B/R} is exact at Ω_{B/R} in the sense that the kernel coincides with the image of the conormal map.

theorem conormalSequence_injective_iff_h1Cotangent_subsingleton (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A] [Algebra.FormallySmooth R A] (B : Type u) [CommRing B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] (hsurj : Function.Surjective ⇑(algebraMap A B)) : Function.Injective ⇑(KaehlerDifferential.kerCotangentToTensor R A B) ↔ Subsingleton (Algebra.H1Cotangent R B)

For a formally smooth R-algebra A and a surjection A → B, the conormal map I/I² → B ⊗_A Ω_{A/R} is injective iff the first cotangent cohomology H¹(L_{B/R}) vanishes.

theorem conormalSequence_injective_of_formallySmooth (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A] [Algebra.FormallySmooth R A] (P : Algebra.Extension R A) [Algebra.FormallySmooth R P.Ring] : Function.Injective ⇑P.cotangentComplex

For a formally smooth R-algebra A with a formally smooth presentation P, the cotangent complex map P.cotangentComplex is injective.

theorem conormalSequence_ses_iff_smooth {X Z S : AlgebraicGeometry.Scheme} (i : Z ⟶ X) [AlgebraicGeometry.IsClosedImmersion i] (f : X ⟶ S) [AlgebraicGeometry.Smooth f] [AlgebraicGeometry.IsNoetherian X] (z : ↥Z) : (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.stalkMap (CategoryTheory.CategoryStruct.comp i f) z)).FormallySmooth ↔ z ∈ AlgebraicGeometry.Scheme.Hom.smoothLocus (CategoryTheory.CategoryStruct.comp i f)

For a closed immersion i : Z → X and a smooth morphism f : X → S, the stalk of i ∘ f at z is formally smooth iff z lies in the smooth locus of i ∘ f.

theorem conormalSequence_smoothLocus_isOpen {X Z S : AlgebraicGeometry.Scheme} (i : Z ⟶ X) [AlgebraicGeometry.IsClosedImmersion i] (f : X ⟶ S) [AlgebraicGeometry.Smooth f] [AlgebraicGeometry.IsNoetherian X] : IsOpen {z : ↥Z | (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.stalkMap (CategoryTheory.CategoryStruct.comp i f) z)).FormallySmooth}

The smooth locus of i ∘ f : Z → S is an open subset of Z.

theorem conormalSequence_smoothness_propagates {X Z S : AlgebraicGeometry.Scheme} (i : Z ⟶ X) [AlgebraicGeometry.IsClosedImmersion i] (f : X ⟶ S) [AlgebraicGeometry.Smooth f] [AlgebraicGeometry.IsNoetherian X] (z₀ : ↥Z) (hz₀ : (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.stalkMap (CategoryTheory.CategoryStruct.comp i f) z₀)).FormallySmooth) : ∃ (U : TopologicalSpace.Opens ↥Z), z₀ ∈ U ∧ ∀ z ∈ U, (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.stalkMap (CategoryTheory.CategoryStruct.comp i f) z)).FormallySmooth

Smoothness propagates to a neighbourhood: a smooth point of i ∘ f has an open neighbourhood of smooth points.