def presheaf_def (C : Type u) [CategoryTheory.Category.{v, u} C] (X : TopCat) : Type (max u v w)

Definition 21 (Lec 10): a presheaf of objects of C on a topological space X is a contravariant functor from Opens X to C.

def pushforward_presheaf_def (C : Type u) [CategoryTheory.Category.{v, u} C] {X Y : TopCat} (f : X ⟶ Y) : CategoryTheory.Functor (TopCat.Presheaf C X) (TopCat.Presheaf C Y)

Definition 23 (Lec 10): the pushforward functor f_* on presheaves induced by a continuous map f : X → Y, sending F to U ↦ F(f⁻¹U).

theorem pushforward_sheaf_of_sheaf {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : TopCat} (f : X ⟶ Y) {F : TopCat.Presheaf C X} (h : F.IsSheaf) : ((TopCat.Presheaf.pushforward C f).obj F).IsSheaf

The pushforward of a sheaf is again a sheaf, so f_* restricts to a functor on sheaves.

@[implicit_reducible]

noncomputable instance presheaf_category_abelian (X : TopCat) : CategoryTheory.Abelian (TopCat.Presheaf AddCommGrpCat X)

Presheaves of abelian groups on X form an abelian category.

@[implicit_reducible]

noncomputable instance sheaf_category_abelian (X : TopCat) : CategoryTheory.Abelian (TopCat.Sheaf AddCommGrpCat X)

Sheaves of abelian groups on X form an abelian category.

instance sheafification_preserves_colimits {C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (D : Type u_1) [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.HasWeakSheafify J D] : CategoryTheory.Limits.PreservesColimitsOfSize.{v, v, max u u_2, max u u_2, max (max (max u u_1) v) u_2, max (max (max u u_1) v) u_2} (CategoryTheory.presheafToSheaf J D)

Proposition 15 (Lec 11): sheafification is a left adjoint and therefore preserves all colimits.

theorem sheafification_iso_of_isSheaf {C : Type u} [CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.HasWeakSheafify J D] {P : CategoryTheory.Functor Cᵒᵖ D} (hP : CategoryTheory.Presheaf.IsSheaf J P) : CategoryTheory.IsIso (CategoryTheory.toSheafify J P)

If P is already a sheaf, then the canonical map P → P# to its sheafification is an isomorphism.

theorem stalk_functor_exact {C : Type v} [CategoryTheory.Category.{u, v} C] [CategoryTheory.Limits.HasColimits C] [CategoryTheory.Limits.HasLimits C] {FC : C → C → Type u_1} {CC : C → Type u} [(X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [CategoryTheory.ConcreteCategory C FC] [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget C)] [CategoryTheory.Abelian C] {X : TopCat} (S : CategoryTheory.ShortComplex (TopCat.Sheaf C X)) : S.Exact ↔ ∀ (x : ↑X), (S.map ((TopCat.Sheaf.forget C X).comp (TopCat.Presheaf.stalkFunctor C x))).Exact

Exactness of a short complex of sheaves can be tested stalkwise: it is exact iff each induced complex of stalks is exact.

instance sheafification_preserves_finite_limits {C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (D : Type u_1) [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.HasSheafify J D] : CategoryTheory.Limits.PreservesFiniteLimits (CategoryTheory.presheafToSheaf J D)

Sheafification is exact: in addition to preserving colimits, it also preserves finite limits.

noncomputable def tilde_gamma_adjunction (R : CommRingCat) : AlgebraicGeometry.tilde.functor R ⊣ AlgebraicGeometry.moduleSpecΓFunctor

The adjunction M ↦ M̃ ⊣ Γ between the tilde construction and global sections on Spec R.

noncomputable def tilde_fullyFaithful (R : CommRingCat) : (AlgebraicGeometry.tilde.functor R).FullyFaithful

The tilde functor M ↦ M̃ from R-modules to quasi-coherent sheaves on Spec R is fully faithful.

instance tilde_isLeftAdjoint_inst (R : CommRingCat) : (AlgebraicGeometry.tilde.functor R).IsLeftAdjoint

Instance: tilde.functor R is a left adjoint (to global sections on Spec R).

class ProjectiveLineBundle (k : Type u) [Field k] : Type (u + 1)

Abstract axiomatization of locally free sheaves (= vector bundles) on P¹_k, packaging rank, twisting sheaves O(d), finite direct sums, and an isomorphism relation. Used to formulate Grothendieck–Birkhoff splitting.

• LocallyFreeSheaf : Type u
• rank : LocallyFreeSheaf k → ℕ
• twistingSheaf : ℤ → LocallyFreeSheaf k
• directSum (ι : Type) [Fintype ι] : (ι → LocallyFreeSheaf k) → LocallyFreeSheaf k
• iso : LocallyFreeSheaf k → LocallyFreeSheaf k → Prop
• twistingSheaf_rank (d : ℤ) : rank (twistingSheaf d) = 1
• iso_refl (E : LocallyFreeSheaf k) : iso E E
• iso_symm (E F : LocallyFreeSheaf k) : iso E F → iso F E
• iso_trans (E F G : LocallyFreeSheaf k) : iso E F → iso F G → iso E G

theorem PLB.rank_one_is_line_bundle (k : Type u) [Field k] [h : ProjectiveLineBundle k] (E : ProjectiveLineBundle.LocallyFreeSheaf k) : ProjectiveLineBundle.rank E = 1 → ∃ (d : ℤ), ProjectiveLineBundle.iso E (ProjectiveLineBundle.twistingSheaf d)

Every line bundle on P¹_k is isomorphic to some twisting sheaf O(d).

theorem PLB.iso_directSum_singleton (k : Type u) [Field k] [h : ProjectiveLineBundle k] (E : ProjectiveLineBundle.LocallyFreeSheaf k) : ProjectiveLineBundle.iso E (ProjectiveLineBundle.directSum (Fin 1) fun (x : Fin 1) => E)

A bundle E is isomorphic to the one-summand direct sum E ≅ ⨁_{i ∈ Fin 1} E.

theorem PLB.iso_directSum_fin_zero (k : Type u) [Field k] [h : ProjectiveLineBundle k] (E : ProjectiveLineBundle.LocallyFreeSheaf k) (f : Fin 0 → ProjectiveLineBundle.LocallyFreeSheaf k) : ProjectiveLineBundle.rank E = 0 → ProjectiveLineBundle.iso E (ProjectiveLineBundle.directSum (Fin 0) f)

A rank-zero bundle is isomorphic to any empty direct sum (the zero bundle).

theorem PLB.split_and_combine (k : Type u) [Field k] [h : ProjectiveLineBundle k] (E : ProjectiveLineBundle.LocallyFreeSheaf k) (n : ℕ) : ProjectiveLineBundle.rank E = n + 2 → (∀ (F : ProjectiveLineBundle.LocallyFreeSheaf k), ProjectiveLineBundle.rank F = n + 1 → ∃ (d : Fin (n + 1) → ℤ), ProjectiveLineBundle.iso F (ProjectiveLineBundle.directSum (Fin (n + 1)) fun (i : Fin (n + 1)) => ProjectiveLineBundle.twistingSheaf (d i))) → ∃ (d : Fin (n + 2) → ℤ), ProjectiveLineBundle.iso E (ProjectiveLineBundle.directSum (Fin (n + 2)) fun (i : Fin (n + 2)) => ProjectiveLineBundle.twistingSheaf (d i))

Inductive step for Grothendieck–Birkhoff: any rank-(n+2) bundle splits off a line bundle and the rank-(n+1) remainder splits into twisting sheaves by hypothesis, giving a splitting of E as a direct sum of O(dᵢ)'s.

theorem PLB.splitting_uniqueness (k : Type u) [Field k] [h : ProjectiveLineBundle k] (n : ℕ) (d d' : Fin n → ℤ) : ProjectiveLineBundle.iso (ProjectiveLineBundle.directSum (Fin n) fun (i : Fin n) => ProjectiveLineBundle.twistingSheaf (d i)) (ProjectiveLineBundle.directSum (Fin n) fun (i : Fin n) => ProjectiveLineBundle.twistingSheaf (d' i)) → ↑(List.ofFn d) = ↑(List.ofFn d')

Uniqueness of Grothendieck–Birkhoff splitting: if ⨁ O(dᵢ) ≅ ⨁ O(dᵢ') then the multisets of degrees {dᵢ} and {dᵢ'} coincide.

theorem gb_existence_aux (k : Type u) [Field k] [h : ProjectiveLineBundle k] (n : ℕ) (E : ProjectiveLineBundle.LocallyFreeSheaf k) : ProjectiveLineBundle.rank E = n → ∃ (d : Fin n → ℤ), ProjectiveLineBundle.iso E (ProjectiveLineBundle.directSum (Fin n) fun (i : Fin n) => ProjectiveLineBundle.twistingSheaf (d i))

Auxiliary induction on rank: every rank-n bundle is isomorphic to a direct sum of twisting sheaves.

theorem grothendieck_birkhoff_splitting (k : Type u) [Field k] [h : ProjectiveLineBundle k] (E : ProjectiveLineBundle.LocallyFreeSheaf k) : ∃ (d : Fin (ProjectiveLineBundle.rank E) → ℤ), ProjectiveLineBundle.iso E (ProjectiveLineBundle.directSum (Fin (ProjectiveLineBundle.rank E)) fun (i : Fin (ProjectiveLineBundle.rank E)) => ProjectiveLineBundle.twistingSheaf (d i))

Grothendieck–Birkhoff splitting theorem (existence): every vector bundle on P¹_k splits as a direct sum of twisting sheaves O(dᵢ).

theorem grothendieck_birkhoff_uniqueness (k : Type u) [Field k] [h : ProjectiveLineBundle k] (E : ProjectiveLineBundle.LocallyFreeSheaf k) (d d' : Fin (ProjectiveLineBundle.rank E) → ℤ) (hd : ProjectiveLineBundle.iso E (ProjectiveLineBundle.directSum (Fin (ProjectiveLineBundle.rank E)) fun (i : Fin (ProjectiveLineBundle.rank E)) => ProjectiveLineBundle.twistingSheaf (d i))) (hd' : ProjectiveLineBundle.iso E (ProjectiveLineBundle.directSum (Fin (ProjectiveLineBundle.rank E)) fun (i : Fin (ProjectiveLineBundle.rank E)) => ProjectiveLineBundle.twistingSheaf (d' i))) : ↑(List.ofFn d) = ↑(List.ofFn d')

Grothendieck–Birkhoff splitting theorem (uniqueness): two splittings of the same bundle E produce the same multiset of degrees.

structure SheafOfModules.LocallyFreeData {C : Type u} [CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [∀ (X : C), (J.over X).HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrpCat)] [∀ (X : C), CategoryTheory.HasWeakSheafify (J.over X) AddCommGrpCat] [∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat] (M : SheafOfModules R) (n : ℕ) : Type (max (u + 1) v)

Witness data that a sheaf of modules M is locally free of rank n (Def 26, Lec 10): a covering family {Xᵢ} and, on each Xᵢ, an isomorphism between M|_{Xᵢ} and the free sheaf of rank n.

• I : Type u
• X : self.I → C
• coversTop : J.CoversTop self.X
• iso (i : self.I) : M.over (self.X i) ≅ free (ULift.{v, 0} (Fin n))

class SheafOfModules.IsLocallyFree {C : Type u} [CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [∀ (X : C), (J.over X).HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrpCat)] [∀ (X : C), CategoryTheory.HasWeakSheafify (J.over X) AddCommGrpCat] [∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat] (M : SheafOfModules R) (n : ℕ) : Prop

A sheaf of modules M is locally free of rank n (Def 26, Lec 10) if local trivialization data exists, i.e. M.LocallyFreeData n is nonempty.

• nonempty_locallyFreeData : Nonempty (M.LocallyFreeData n)

def AlgebraicGeometry.Scheme.IsLocallyFreeSheaf (X : Scheme) (F : X.Modules) (n : ℕ) : Prop

Scheme-theoretic version of locally free: there exists an open cover of X on each piece of which F is isomorphic to the free O_X-module of rank n.