noncomputable def grassmannianTracePairing (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) [FiniteDimensional F ↥V.toSubmodule] : grassmannianTangentSpace F W kk V →ₗ[F] grassmannianCotangentSpace F W kk V →ₗ[F] F

The trace pairing on the Grassmannian: a k-bilinear pairing T_V Gr × T^*_V Gr → F, sending (φ, ψ) ↦ tr(ψ ∘ φ).

noncomputable def cotangentToDualTangent (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (kk : ℕ) (V : Module.Grassmannian F W kk) [FiniteDimensional F ↥V.toSubmodule] : grassmannianCotangentSpace F W kk V →ₗ[F] Module.Dual F (grassmannianTangentSpace F W kk V)

The natural map T^*_V Gr → (T_V Gr)* induced by the trace pairing.

theorem prop37_cotangent_dim_explicit (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 (grassmannianCotangentSpace F W kk V) = kk * (Module.finrank F W - kk)

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

theorem prop37_tangent_dim_explicit (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 (grassmannianTangentSpace F W kk V) = (Module.finrank F W - kk) * kk

Proposition 37 (tangent form). The tangent space Hom(V, W/V) at a point V ∈ Gr(k, W) has dimension (dim W - k) · k.

theorem prop37_tangent_cotangent_dim_eq (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 (grassmannianTangentSpace F W kk V) = Module.finrank F (grassmannianCotangentSpace F W kk V)

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

theorem prop37_grassmannian_dimension (F : Type u) [Field F] (W : Type v) [AddCommGroup W] [Module F W] (n kk : ℕ) [FiniteDimensional F W] (hW : Module.finrank F W = n) (V : Module.Grassmannian F W kk) : Module.finrank F (grassmannianCotangentSpace F W kk V) = kk * (n - kk)

Dimension of the Grassmannian Gr(k, n) via its cotangent space: k · (n - k).

theorem prop37_projective_cotangent_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 (grassmannianCotangentSpace F W 1 V) = n

Specialisation to projective space: the cotangent space at a point of ℙⁿ has dimension n.