noncomputable def Lec20TangentCone.scaleAlgHom {R : Type u} [CommRing R] (c : R) : Polynomial R →ₐ[R] Polynomial R

The R-algebra homomorphism R[X] → R[X] sending X ↦ c · X, used to realise the scalar R*-action on the Rees algebra and hence on the tangent cone.

theorem Lec20TangentCone.scaleAlgHom_one {R : Type u} [CommRing R] : scaleAlgHom 1 = AlgHom.id R (Polynomial R)

Scaling by 1 on R[X] is the identity homomorphism.

theorem Lec20TangentCone.scaleAlgHom_comp {R : Type u} [CommRing R] (c₁ c₂ : R) : (scaleAlgHom c₁).comp (scaleAlgHom c₂) = scaleAlgHom (c₁ * c₂)

Composition of scaling homomorphisms corresponds to multiplication: scaling by c₁ after scaling by c₂ equals scaling by c₁ * c₂.

theorem Lec20TangentCone.scaleAlgHom_mem_reesAlgebra {R : Type u} [CommRing R] (I : Ideal R) (c : R) (f : Polynomial R) (hf : f ∈ reesAlgebra I) : (scaleAlgHom c) f ∈ reesAlgebra I

The scaling endomorphism scaleAlgHom c preserves the Rees subalgebra of R[X] associated to an ideal I.

noncomputable def Lec20TangentCone.scaleOnRees {R : Type u} [CommRing R] (I : Ideal R) (c : R) : ↥(reesAlgebra I) →+* ↥(reesAlgebra I)

The ring endomorphism of the Rees algebra reesAlgebra I induced by scaling X ↦ c · X.

theorem Lec20TangentCone.scaleOnRees_algebraMap {R : Type u} [CommRing R] (I : Ideal R) (c r : R) : (scaleOnRees I c) ((algebraMap R ↥(reesAlgebra I)) r) = (algebraMap R ↥(reesAlgebra I)) r

The scaling endomorphism of the Rees algebra fixes the image of R under the structure map.

noncomputable def Lec20TangentCone.reesIdealFromBase {R : Type u} [CommRing R] (𝔪 : Ideal R) : Ideal ↥(reesAlgebra 𝔪)

The ideal of the Rees algebra obtained by extending an ideal 𝔪 ⊆ R along the structure map; its quotient defines the associated graded ring.

theorem Lec20TangentCone.reesIdealFromBase_le_comap_scaleOnRees {R : Type u} [CommRing R] (I : Ideal R) (c : R) : reesIdealFromBase I ≤ Ideal.comap (scaleOnRees I c) (reesIdealFromBase I)

The extended ideal reesIdealFromBase I is preserved (in the comap sense) by every scaling endomorphism of the Rees algebra.

def Lec20TangentCone.assocGradedRing {R : Type u} [CommRing R] (𝔪 : Ideal R) : Type u

The associated graded ring gr_𝔪(R) = ⨁ 𝔪^n/𝔪^{n+1}, realised as the quotient of the Rees algebra by the ideal generated by 𝔪.

@[implicit_reducible]

noncomputable instance Lec20TangentCone.assocGradedRing.commRing {R : Type u} [CommRing R] (𝔪 : Ideal R) : CommRing (assocGradedRing 𝔪)

The commutative ring structure on the associated graded ring, inherited from the quotient construction.

noncomputable def Lec20TangentCone.scaleOnGraded {R : Type u} [CommRing R] (I : Ideal R) (c : R) : assocGradedRing I →+* assocGradedRing I

The induced ring endomorphism of the associated graded ring gr_I(R) coming from the R*-scaling action on the Rees algebra.

theorem Lec20TangentCone.tangentCone_isCone_identity {R : Type u} [CommRing R] (I : Ideal R) : scaleOnGraded I 1 = RingHom.id (assocGradedRing I)

The tangent cone is a cone: scaling by 1 acts as the identity on the associated graded ring.

theorem Lec20TangentCone.tangentCone_isCone_compose {R : Type u} [CommRing R] (I : Ideal R) (c₁ c₂ : R) : scaleOnGraded I (c₁ * c₂) = (scaleOnGraded I c₁).comp (scaleOnGraded I c₂)

The tangent cone is a cone: the scaling action is multiplicative, (c₁ * c₂)·_ = c₁ ·(c₂ ·_) on gr_I(R).

noncomputable def Lec20TangentCone.ringHomToAssocGraded {R : Type u} [CommRing R] (𝔪 : Ideal R) : R →+* assocGradedRing 𝔪

The canonical ring homomorphism R → gr_𝔪(R), factoring through the structure map into the Rees algebra and the quotient projection.

theorem Lec20TangentCone.tangentCone_over_closedPoint {R : Type u} [CommRing R] (𝔪 : Ideal R) : 𝔪 ≤ RingHom.ker (ringHomToAssocGraded 𝔪)

The tangent cone sits over the closed point: the ideal 𝔪 lies in the kernel of R → gr_𝔪(R), so the map factors through the residue field R⧸𝔪.

noncomputable def Lec20TangentCone.tangentCone {R : Type u} [CommRing R] (𝔪 : Ideal R) : AlgebraicGeometry.Scheme

The tangent cone C_x X := Spec(gr_𝔪 O_{X,x}) of X = Spec R at the point cut out by 𝔪.

@[reducible, inline]

noncomputable abbrev Lec20TangentCone.definition_38_tangentCone {R : Type u} [CommRing R] (𝔪 : Ideal R) : AlgebraicGeometry.Scheme

Definition 38: the tangent cone of Spec R at a point with maximal ideal 𝔪, defined as Spec(gr_𝔪(R)).

noncomputable def Lec20TangentCone.tangentConeMap {R : Type u} [CommRing R] (𝔪 : Ideal R) : tangentCone 𝔪 ⟶ AlgebraicGeometry.Spec (CommRingCat.of R)

The structure morphism C_x X → X = Spec R from the tangent cone to the ambient scheme.

@[implicit_reducible]

noncomputable instance Lec20TangentCone.assocGradedRing.algebraQuot {R : Type u} [CommRing R] (𝔪 : Ideal R) : Algebra (R ⧸ 𝔪) (assocGradedRing 𝔪)

The associated graded ring is an algebra over the residue field R ⧸ 𝔪.

@[implicit_reducible]

noncomputable instance Lec20TangentCone.assocGradedRing.algebraRees {R : Type u} [CommRing R] (𝔪 : Ideal R) : Algebra (↥(reesAlgebra 𝔪)) (assocGradedRing 𝔪)

The associated graded ring is an algebra over the Rees algebra via the quotient map.

noncomputable def Lec20TangentCone.assocGradedRing_equivExceptionalFiber {R : Type u} [CommRing R] (𝔪 : Ideal R) : assocGradedRing 𝔪 ≃ₐ[↥(reesAlgebra 𝔪)] TensorProduct R (↥(reesAlgebra 𝔪)) (R ⧸ 𝔪)

Algebra isomorphism realising the associated graded ring as the exceptional fiber of the blow-up: gr_𝔪(R) ≃ reesAlgebra 𝔪 ⊗_R (R⧸𝔪).

noncomputable def Lec20TangentCone.assocGradedRing_ringEquivExceptionalFiber {R : Type u} [CommRing R] (𝔪 : Ideal R) : assocGradedRing 𝔪 ≃+* TensorProduct R (↥(reesAlgebra 𝔪)) (R ⧸ 𝔪)

The ring isomorphism underlying the algebra equivalence gr_𝔪(R) ≃ reesAlgebra 𝔪 ⊗_R (R⧸𝔪).

noncomputable def Lec20TangentCone.exceptionalFiberScheme {R : Type u} [CommRing R] (𝔪 : Ideal R) : AlgebraicGeometry.Scheme

The exceptional fiber of the blow-up Bl_𝔪(Spec R) → Spec R above the closed point, realised as Spec(reesAlgebra 𝔪 ⊗_R (R⧸𝔪)).

noncomputable def Lec20TangentCone.tangentCone_isoExceptionalFiber {R : Type u} [CommRing R] (𝔪 : Ideal R) : tangentCone 𝔪 ≅ exceptionalFiberScheme 𝔪

Scheme-theoretic isomorphism identifying the tangent cone with the exceptional fiber of the blow-up.

def Lec20TangentCone.assocGradedPiece {R : Type u} [CommRing R] (𝔪 : Ideal R) (n : ℕ) : Type u

The n-th graded piece 𝔪^n / 𝔪^{n+1} of the associated graded ring gr_𝔪(R).

@[implicit_reducible]

instance Lec20TangentCone.assocGradedPiece.addCommGroup {R : Type u} [CommRing R] (𝔪 : Ideal R) (n : ℕ) : AddCommGroup (assocGradedPiece 𝔪 n)

The additive group structure on the graded piece 𝔪^n / 𝔪^{n+1}.

@[implicit_reducible]

instance Lec20TangentCone.assocGradedPiece.module {R : Type u} [CommRing R] (𝔪 : Ideal R) (n : ℕ) : Module R (assocGradedPiece 𝔪 n)

The R-module structure on the graded piece 𝔪^n / 𝔪^{n+1}.

noncomputable def Lec20TangentCone.reesToAssocGraded {R : Type u} [CommRing R] (𝔪 : Ideal R) : ↥(reesAlgebra 𝔪) →+* assocGradedRing 𝔪

The quotient ring homomorphism from the Rees algebra onto the associated graded ring.

theorem Lec20TangentCone.reesToAssocGraded_surjective {R : Type u} [CommRing R] (𝔪 : Ideal R) : Function.Surjective ⇑(reesToAssocGraded 𝔪)

The projection reesAlgebra 𝔪 → gr_𝔪(R) is surjective.

theorem Lec20TangentCone.proposition_38_smooth_graded_iso_symmetric (R : Type u) [CommRing R] [IsLocalRing R] [IsRegularLocalRing R] : Nonempty (assocGradedRing (IsLocalRing.maximalIdeal R) ≃+* SymmetricAlgebra (IsLocalRing.ResidueField R) (IsLocalRing.CotangentSpace R))

Proposition 38 (smooth case): At a smooth (regular) point, the associated graded ring is isomorphic to the symmetric algebra on the cotangent space, so the tangent cone is the linear tangent space Sym(T*_x X).

theorem Lec20TangentCone.proposition_38 {R : Type u} [CommRing R] (𝔪 : Ideal R) : Nonempty (tangentCone 𝔪 ≅ exceptionalFiberScheme 𝔪) ∧ ∀ (S : Type u) [inst : CommRing S] [inst_1 : IsLocalRing S] [IsRegularLocalRing S], Nonempty (assocGradedRing (IsLocalRing.maximalIdeal S) ≃+* SymmetricAlgebra (IsLocalRing.ResidueField S) (IsLocalRing.CotangentSpace S))

Proposition 38: The tangent cone of Spec R at 𝔪 is the exceptional fiber of the blow-up, and at a regular (smooth) local ring it coincides with Sym(T*_x X).

@[reducible, inline]

abbrev Lec20TangentCone.tangentSpaceRing (R : Type u) [CommRing R] [IsLocalRing R] : Type u

The coordinate ring of the linear tangent space at the closed point of a local ring, namely the symmetric algebra on the cotangent space.

noncomputable def Lec20TangentCone.tangentSpaceScheme (R : Type u) [CommRing R] [IsLocalRing R] : AlgebraicGeometry.Scheme

The (Zariski) tangent space scheme Spec(Sym(T*_x X)) at the closed point.

theorem Lec20TangentCone.symToAssocGradedSurjection (R : Type u) [CommRing R] [IsLocalRing R] : ∃ (f : tangentSpaceRing R →+* assocGradedRing (IsLocalRing.maximalIdeal R)), Function.Surjective ⇑f

There is a canonical surjective ring homomorphism from the symmetric algebra on the cotangent space onto the associated graded ring; equivalently, the tangent cone embeds into the linear tangent space.