theorem SmoothClosedSubvariety.corollary25_smooth_closed_subvariety_criterion (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) [Algebra.FormallySmooth k R] : Algebra.FormallySmooth k (R ⧸ I) ↔ ∃ (l : TensorProduct R (R ⧸ I) Ω[R⁄k] →ₗ[R] (RingHom.ker (algebraMap R (R ⧸ I))).Cotangent), l ∘ₗ KaehlerDifferential.kerCotangentToTensor k R (R ⧸ I) = LinearMap.id

Corollary 25 of Lecture 20: For a formally smooth k-algebra R, the quotient R ⧸ I is formally smooth iff the kernel-to-cotangent map kerCotangentToTensor k R (R ⧸ I) splits.

theorem SmoothClosedSubvariety.proposition35_conormal_right_exact (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) : Function.Exact ⇑(KaehlerDifferential.kerCotangentToTensor k R (R ⧸ I)) ⇑(KaehlerDifferential.mapBaseChange k R (R ⧸ I))

Proposition 35 (right exactness): The conormal sequence I/I² → (R⧸I) ⊗_R Ω[R/k] → Ω[(R⧸I)/k] → 0 is exact at the middle term.

theorem SmoothClosedSubvariety.proposition35_conormal_surjective (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) : Function.Surjective ⇑(KaehlerDifferential.mapBaseChange k R (R ⧸ I))

Proposition 35 (surjectivity): The base-change map (R⧸I) ⊗_R Ω[R/k] → Ω[(R⧸I)/k] is surjective.

theorem SmoothClosedSubvariety.corollary26_conormal_ses_injective (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) [Algebra.FormallySmooth k R] [Algebra.FormallySmooth k (R ⧸ I)] : Function.Injective ⇑(KaehlerDifferential.kerCotangentToTensor k R (R ⧸ I))

Corollary 26 (injectivity): When both R and R ⧸ I are formally smooth over k, the conormal map I/I² → (R⧸I) ⊗_R Ω[R/k] is injective.

theorem SmoothClosedSubvariety.corollary26_conormal_ses_exact (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) : Function.Exact ⇑(KaehlerDifferential.kerCotangentToTensor k R (R ⧸ I)) ⇑(KaehlerDifferential.mapBaseChange k R (R ⧸ I))

Corollary 26 (exactness): The conormal sequence is exact at (R⧸I) ⊗_R Ω[R/k].

theorem SmoothClosedSubvariety.corollary26_conormal_ses_surjective (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) : Function.Surjective ⇑(KaehlerDifferential.mapBaseChange k R (R ⧸ I))

Corollary 26 (surjectivity): The base-change map in the conormal sequence is surjective.

theorem SmoothClosedSubvariety.corollary26_conormal_ses_split (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) [Algebra.FormallySmooth k R] [Algebra.FormallySmooth k (R ⧸ I)] : ∃ (l : TensorProduct R (R ⧸ I) Ω[R⁄k] →ₗ[R] (RingHom.ker (algebraMap R (R ⧸ I))).Cotangent), l ∘ₗ KaehlerDifferential.kerCotangentToTensor k R (R ⧸ I) = LinearMap.id

Corollary 26 (splitting): When both R and R ⧸ I are formally smooth over k, the conormal short exact sequence splits.

theorem SmoothClosedSubvariety.corollary26_conormal_locally_free_direct_summand (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) [Algebra.FormallySmooth k R] [Algebra.FormallySmooth k (R ⧸ I)] : ∃ (l : TensorProduct R (R ⧸ I) Ω[R⁄k] →ₗ[R] (RingHom.ker (algebraMap R (R ⧸ I))).Cotangent), l ∘ₗ KaehlerDifferential.kerCotangentToTensor k R (R ⧸ I) = LinearMap.id ∧ Function.Injective ⇑(KaehlerDifferential.kerCotangentToTensor k R (R ⧸ I))

Corollary 26 (direct summand form): In the smooth case, I/I² is a locally free direct summand of (R⧸I) ⊗_R Ω[R/k], realised by an injective map that admits a left inverse.

theorem SmoothClosedSubvariety.corollary26_canonical_bundle_formula (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) [Algebra.FormallySmooth k R] [Algebra.FormallySmooth k (R ⧸ I)] [Nontrivial (R ⧸ I)] [Module.Free (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k])] [Module.Finite (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k])] [Module.Free (R ⧸ I) Ω[R ⧸ I⁄k]] [Module.Finite (R ⧸ I) Ω[R ⧸ I⁄k]] [Module.Free (R ⧸ I) I.Cotangent] [Module.Finite (R ⧸ I) I.Cotangent] (d c : ℕ) (hd : Module.finrank (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k]) = d) (hc : Module.finrank (R ⧸ I) I.Cotangent = c) (hdc : Module.finrank (R ⧸ I) Ω[R ⧸ I⁄k] = d - c) : Nonempty (↥(⋀[R ⧸ I]^d (TensorProduct R (R ⧸ I) Ω[R⁄k])) ≃ₗ[R ⧸ I] TensorProduct (R ⧸ I) ↥(⋀[R ⧸ I]^(d - c) Ω[R ⧸ I⁄k]) ↥(⋀[R ⧸ I]^c I.Cotangent))

Corollary 26 (canonical bundle formula): For a smooth closed subvariety of codimension c in a smooth d-dimensional variety, the top exterior power of the ambient cotangent bundle restricted to Z decomposes as the top wedge of Ω[Z/k] tensor the wedge of I/I².

theorem SmoothClosedSubvariety.corollary26_adjunction_formula (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) [Algebra.FormallySmooth k R] [Algebra.FormallySmooth k (R ⧸ I)] [Nontrivial (R ⧸ I)] [Module.Free (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k])] [Module.Finite (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k])] [Module.Free (R ⧸ I) Ω[R ⧸ I⁄k]] [Module.Finite (R ⧸ I) Ω[R ⧸ I⁄k]] [Module.Free (R ⧸ I) I.Cotangent] [Module.Finite (R ⧸ I) I.Cotangent] (d : ℕ) (hd : Module.finrank (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k]) = d) (hc : Module.finrank (R ⧸ I) I.Cotangent = 1) (hdc : Module.finrank (R ⧸ I) Ω[R ⧸ I⁄k] = d - 1) : Nonempty (↥(⋀[R ⧸ I]^d (TensorProduct R (R ⧸ I) Ω[R⁄k])) ≃ₗ[R ⧸ I] TensorProduct (R ⧸ I) ↥(⋀[R ⧸ I]^(d - 1) Ω[R ⧸ I⁄k]) ↥(⋀[R ⧸ I]^1 I.Cotangent))

Corollary 26 (adjunction formula): The divisor case (c = 1) of the canonical bundle formula relating the determinant of the cotangent bundle on Z to the conormal line bundle.

@[reducible, inline]

abbrev SmoothClosedSubvariety.NormalModule (R : Type u_3) [CommRing R] (I : Ideal R) : Type u_3

The normal module N_{Z/X} = Hom_{R⧸I}(I/I², R⧸I), dual to the conormal module I/I².

theorem SmoothClosedSubvariety.adjunction_formula_with_twisting (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) [Algebra.FormallySmooth k R] [Algebra.FormallySmooth k (R ⧸ I)] [Nontrivial (R ⧸ I)] [Module.Free (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k])] [Module.Finite (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k])] [Module.Free (R ⧸ I) Ω[R ⧸ I⁄k]] [Module.Finite (R ⧸ I) Ω[R ⧸ I⁄k]] [Module.Free (R ⧸ I) I.Cotangent] [Module.Finite (R ⧸ I) I.Cotangent] (d c : ℕ) (hd : Module.finrank (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k]) = d) (hc : Module.finrank (R ⧸ I) I.Cotangent = c) (hdc : Module.finrank (R ⧸ I) Ω[R ⧸ I⁄k] = d - c) : Nonempty (↥(⋀[R ⧸ I]^(d - c) Ω[R ⧸ I⁄k]) ≃ₗ[R ⧸ I] TensorProduct (R ⧸ I) ↥(⋀[R ⧸ I]^d (TensorProduct R (R ⧸ I) Ω[R⁄k])) ↥(⋀[R ⧸ I]^c (NormalModule R I)))

Adjunction formula with twisting: rewrites the canonical bundle of the subvariety as the restriction of the ambient canonical bundle twisted by the wedge of the normal bundle.

theorem SmoothClosedSubvariety.adjunction_formula_divisor_twisting (k : Type u_1) [Field k] (R : Type u_2) [CommRing R] [Algebra k R] (I : Ideal R) [Algebra.FormallySmooth k R] [Algebra.FormallySmooth k (R ⧸ I)] [Nontrivial (R ⧸ I)] [Module.Free (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k])] [Module.Finite (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k])] [Module.Free (R ⧸ I) Ω[R ⧸ I⁄k]] [Module.Finite (R ⧸ I) Ω[R ⧸ I⁄k]] [Module.Free (R ⧸ I) I.Cotangent] [Module.Finite (R ⧸ I) I.Cotangent] (d : ℕ) (hd : Module.finrank (R ⧸ I) (TensorProduct R (R ⧸ I) Ω[R⁄k]) = d) (hc : Module.finrank (R ⧸ I) I.Cotangent = 1) (hdc : Module.finrank (R ⧸ I) Ω[R ⧸ I⁄k] = d - 1) : Nonempty (↥(⋀[R ⧸ I]^(d - 1) Ω[R ⧸ I⁄k]) ≃ₗ[R ⧸ I] TensorProduct (R ⧸ I) ↥(⋀[R ⧸ I]^d (TensorProduct R (R ⧸ I) Ω[R⁄k])) ↥(⋀[R ⧸ I]^1 (NormalModule R I)))

Adjunction formula with twisting for divisors: the special case c = 1 expressing the canonical bundle of a smooth divisor in terms of the ambient canonical bundle and the normal line bundle.