Atlas.AlgebraicGeometryI.code.Lec20GrassmannianTangent

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theorem prop35a_smooth_kahler_projective (R : Type u) [CommRing R] (A : Type v) [CommRing A] [Algebra R A] [Algebra.FormallySmooth R A] :

Module.Projective A Ω[A⁄R]

Proposition 35(a). For a formally smooth R-algebra A, the module of Kähler differentials Ω_{A/R} is projective.

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theorem prop35b_conormal_exact (R : Type u) [CommRing R] (A : Type v) [CommRing A] [Algebra R A] (B : Type w) [CommRing B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] (hAB : Function.Surjective ⇑(algebraMap A B)) :

(KaehlerDifferential.kerCotangentToTensor R A B).range = Submodule.restrictScalars A (KaehlerDifferential.mapBaseChange R A B).ker

Proposition 35(b) — exactness of the conormal sequence at the middle term: for a surjection A → B, the image of I/I² → B ⊗_A Ω_{A/R} is the kernel of the map to Ω_{B/R}.

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theorem prop35b_conormal_surjective (R : Type u) [CommRing R] (A : Type v) [CommRing A] [Algebra R A] (B : Type w) [CommRing B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] :

Function.Surjective ⇑(KaehlerDifferential.map R A B B)

Proposition 35(b) — the map Ω_{A/R} ⊗_A B → Ω_{B/R} is surjective.

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theorem prop35b_jacobi_zariski_exact (R : Type u) [CommRing R] (A : Type v) [CommRing A] [Algebra R A] (B : Type w) [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)

Proposition 35(b) — exactness of the Jacobi-Zariski sequence B ⊗_A Ω_{A/R} → Ω_{B/R} at Ω_{B/R}.

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theorem prop35c_smooth_conormal_injective (R : Type u) [CommRing R] (A : Type v) [CommRing A] [Algebra R A] [Algebra.FormallySmooth R A] (P : Algebra.Extension R A) [Algebra.FormallySmooth R P.Ring] :

Function.Injective ⇑P.cotangentComplex

Proposition 35(c). For a formally smooth R-algebra A with a formally smooth presentation P, the conormal map P.cotangentComplex is injective.

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theorem cor25_smooth_criterion (R : Type u) [CommRing R] (B : Type w) [CommRing B] [Algebra R B] :

Algebra.FormallySmooth R B ↔ Module.Projective B Ω[B⁄R] ∧ Subsingleton (Algebra.H1Cotangent R B)

Corollary 25 (Lecture 20). A R-algebra B is formally smooth iff Ω_{B/R} is projective over B and the first cotangent cohomology H¹(L_{B/R}) vanishes.

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theorem cor25_conormal_injective_iff_h1_vanishes (R : Type u) [CommRing R] (A : Type v) [CommRing A] [Algebra R A] [Algebra.FormallySmooth R A] (B : Type w) [CommRing B] [Algebra R B] [Algebra A B] [IsScalarTower R A B] (hAB : Function.Surjective ⇑(algebraMap A B)) :

Function.Injective ⇑(KaehlerDifferential.kerCotangentToTensor R A B) ↔ Subsingleton (Algebra.H1Cotangent R B)

Corollary 25 — conormal injectivity: for a formally smooth A and surjection A → B, the conormal map is injective iff H¹(L_{B/R}) vanishes.

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@[reducible, inline]

abbrev PolyRingLec20 (k : Type u) [Field k] (n : ℕ) :

Type u

Shorthand for the polynomial ring k[x_0, …, x_n] used in the Euler sequence section.

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noncomputable def euler_seq_kahler_basis (k : Type u) [Field k] (n : ℕ) :

Module.Basis (Fin (n + 1)) (PolyRingLec20 k n) Ω[PolyRingLec20 k n⁄k]

The standard basis (dx_0, …, dx_n) of Ω_{k[x_0,…,x_n]/k} as a free module of rank n + 1.

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theorem euler_seq_kahler_rank (k : Type u) [Field k] (n : ℕ) :

Module.finrank (PolyRingLec20 k n) Ω[PolyRingLec20 k n⁄k] = n + 1

The module of Kähler differentials of k[x_0,…,x_n] has rank n + 1.

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instance euler_seq_kahler_free (k : Type u) [Field k] (n : ℕ) :

Module.Free (PolyRingLec20 k n) Ω[PolyRingLec20 k n⁄k]

The module of Kähler differentials of k[x_0,…,x_n] is free.

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instance euler_seq_kahler_finite (k : Type u) [Field k] (n : ℕ) :

Module.Finite (PolyRingLec20 k n) Ω[PolyRingLec20 k n⁄k]

The module of Kähler differentials of k[x_0,…,x_n] is finitely generated.

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theorem euler_seq_canonical_rank (k : Type u) [Field k] (n : ℕ) :

Module.finrank (PolyRingLec20 k n) ↥(⋀[PolyRingLec20 k n]^(n + 1) Ω[PolyRingLec20 k n⁄k]) = 1

The top exterior power ∧^{n+1} Ω of the Kähler differentials of k[x_0,…,x_n] is free of rank one (the canonical module on affine (n+1)-space).

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def tautologicalSubspace (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) :

Submodule F W

The tautological subspace V ⊂ W associated to a point V ∈ Gr(k, W).

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@[simp]

theorem tautologicalSubspace_eq (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) :

tautologicalSubspace F W kk V = V.toSubmodule

The tautological subspace is, by definition, just V.toSubmodule.

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@[reducible, inline]

abbrev tautologicalQuotient (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) :

Type v

The tautological quotient bundle fibre W/V at a point V ∈ Gr(k, W).

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instance tautologicalQuotient_finite (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) :

Module.Finite F (tautologicalQuotient F W kk V)

The tautological quotient W/V is a finite F-module.

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instance tautologicalQuotient_projective (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) :

Module.Projective F (tautologicalQuotient F W kk V)

The tautological quotient W/V is a projective F-module.

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theorem tautological_rank_constant (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) (p : PrimeSpectrum F) :

Module.rankAtStalk (W ⧸ V.toSubmodule) p = kk

The tautological quotient bundle has constant rank k at every prime of the base.

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theorem grassmannian_quotient_finrank (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) :

Module.finrank F (W ⧸ V.toSubmodule) = kk

The quotient W/V at a point V ∈ Gr(k, W) has dimension k.

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theorem grassmannian_submodule_finrank (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] [FiniteDimensional F W] (kk : ℕ) (V : Module.Grassmannian F W kk) :

Module.finrank F ↥V.toSubmodule = Module.finrank F W - kk

The dimension of V ⊂ W at a point V ∈ Gr(k, W) is dim W - k.

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@[reducible, inline]

abbrev grassmannianTangentSpace (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) :

Type v

Tangent space of the Grassmannian at a point V, identified with Hom(V, W/V).

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@[reducible, inline]

abbrev grassmannianCotangentSpace (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) :

Type v

Cotangent space of the Grassmannian at a point V, identified with Hom(W/V, V).

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theorem prop37_grassmannian_tangent_dim (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] [FiniteDimensional F W] (kk : ℕ) (V : Module.Grassmannian F W kk) [FiniteDimensional F ↥V.toSubmodule] [FiniteDimensional F (W ⧸ V.toSubmodule)] :

Module.finrank F (grassmannianTangentSpace F W kk V) = Module.finrank F ↥V.toSubmodule * Module.finrank F (W ⧸ V.toSubmodule)

Proposition 37 (Lecture 20). The tangent space Hom(V, W/V) of Gr(k, W) at V has dimension dim V · dim(W/V).

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theorem prop37_grassmannian_cotangent_dim (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] [FiniteDimensional F W] (kk : ℕ) (V : Module.Grassmannian F W kk) [FiniteDimensional F ↥V.toSubmodule] [FiniteDimensional F (W ⧸ V.toSubmodule)] :

Module.finrank F (grassmannianCotangentSpace F W kk V) = Module.finrank F (W ⧸ V.toSubmodule) * Module.finrank F ↥V.toSubmodule

Proposition 37 (Lecture 20). The cotangent space Hom(W/V, V) of Gr(k, W) at V has dimension dim(W/V) · dim V.

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theorem prop37_tangent_eq_cotangent_dim (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] [FiniteDimensional F W] (kk : ℕ) (V : Module.Grassmannian F W kk) [FiniteDimensional F ↥V.toSubmodule] [FiniteDimensional F (W ⧸ V.toSubmodule)] :

Module.finrank F (grassmannianTangentSpace F W kk V) = Module.finrank F (grassmannianCotangentSpace F W kk V)

The tangent and cotangent spaces of the Grassmannian have equal dimension.

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theorem prop37_euler_specialization_dim (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] [FiniteDimensional F W] (V : Module.Grassmannian F W 1) [FiniteDimensional F ↥V.toSubmodule] [FiniteDimensional F (W ⧸ V.toSubmodule)] :

Module.finrank F (grassmannianTangentSpace F W 1 V) = Module.finrank F ↥V.toSubmodule * Module.finrank F (W ⧸ V.toSubmodule)

Specialisation to Gr(1, W) = ℙ(W): the tangent space at V has dimension dim V · dim(W/V).

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theorem prop37_projective_space_dim (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (n : ℕ) [FiniteDimensional F W] (hW : Module.finrank F W = n + 1) (V : Module.Grassmannian F W 1) :

Module.finrank F (grassmannianTangentSpace F W 1 V) = n

The tangent space of ℙⁿ at every point has dimension n.