def IsQuasicoherentSheaf {X : AlgebraicGeometry.Scheme} (ℱ : X.Modules) : Prop

A sheaf of O_X-modules ℱ is quasi-coherent (Def 24, Lec 10) — abbreviation for the mathlib predicate SheafOfModules.IsQuasicoherent.

noncomputable def IsCoherentSheaf {X : AlgebraicGeometry.Scheme} : X.Modules → Prop

Predicate for being a coherent sheaf of O_X-modules (Lec 10, Def 25 + finite type). Placeholder — to be specialized to mathlib's coherence notion.

structure GrMod {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : Type (u + 1)

A graded module over a graded ring 𝒜 = ⨁ₙ 𝒜ₙ: an A-module carrier together with a ℤ-indexed decomposition carrier = ⨁_d component d compatible with the grading in the sense 𝒜ₙ · component d ⊆ component (n + d).

• carrier : Type u
• instAddCommGroup : AddCommGroup self.carrier
• instModule : Module A self.carrier
• component : ℤ → AddSubgroup self.carrier
• internal : DirectSum.IsInternal self.component
• graded_smul (n : ℕ) (d : ℤ) (a : A) (m : self.carrier) : a ∈ ↑(𝒜 n) → m ∈ self.component d → a • m ∈ self.component (↑n + d)

def GrMod.IsFinitelyGenerated {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] {𝒜 : ℕ → σ} [GradedRing 𝒜] (M : GrMod 𝒜) : Prop

A graded module is finitely generated if its underlying A-module is finite.

def GrMod.IsFiniteDimensional {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] {𝒜 : ℕ → σ} [GradedRing 𝒜] (M : GrMod 𝒜) : Prop

A graded module is "finite-dimensional" (concentrated in finitely many degrees) if all but finitely many graded components vanish.

def GrMod.IsLocallyNilpotent {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] {𝒜 : ℕ → σ} [GradedRing 𝒜] (M : GrMod 𝒜) : Prop

A graded module is locally nilpotent with respect to the irrelevant ideal 𝒜₊ if every element is annihilated by a power of every element of 𝒜₊.

structure GrMod.Hom {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] {𝒜 : ℕ → σ} [GradedRing 𝒜] (M N : GrMod 𝒜) : Type u

A morphism of graded modules: an underlying A-linear map (here without graded constraints — kept as a flexible wrapper).

• toLinearMap : M.carrier →ₗ[A] N.carrier

theorem GrMod.Hom.ext {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] {𝒜 : ℕ → σ} [GradedRing 𝒜] {M N : GrMod 𝒜} {f g : M.Hom N} (h : f.toLinearMap = g.toLinearMap) : f = g

Extensionality for GrMod.Hom: two morphisms are equal iff their underlying linear maps coincide.

theorem GrMod.Hom.ext_iff {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] {𝒜 : ℕ → σ} [GradedRing 𝒜] {M N : GrMod 𝒜} {f g : M.Hom N} : f = g ↔ f.toLinearMap = g.toLinearMap

structure GrMod.ShortExact {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] {𝒜 : ℕ → σ} [GradedRing 𝒜] (M₁ M₂ M₃ : GrMod 𝒜) : Type u

A short exact sequence of graded modules 0 → M₁ → M₂ → M₃ → 0, packaged with injectivity of f, surjectivity of g, and exactness ker g = range f.

• f : M₁.carrier →ₗ[A] M₂.carrier
• g : M₂.carrier →ₗ[A] M₃.carrier
• f_injective : Function.Injective ⇑self.f
• g_surjective : Function.Surjective ⇑self.g
• exact : self.g.ker = self.f.range

noncomputable def GrMod.shift {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (d : ℤ) : GrMod 𝒜

Shift A(d) of the graded ring viewed as a graded module: the module A with grading translated by d.

noncomputable def GrMod.directSumCopies {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] {𝒜 : ℕ → σ} [GradedRing 𝒜] (M : GrMod 𝒜) (k : ℕ) : GrMod 𝒜

Direct sum of k copies of a graded module M, again a graded module.

noncomputable def GrMod.kernelGrMod {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] {𝒜 : ℕ → σ} [GradedRing 𝒜] (M N : GrMod 𝒜) (f : M.carrier →ₗ[A] N.carrier) : GrMod 𝒜

Kernel of an A-linear map between graded modules, packaged as a graded module.

noncomputable def GrMod.shortExactOfSurjection {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] {𝒜 : ℕ → σ} [GradedRing 𝒜] (M N : GrMod 𝒜) (f : M.carrier →ₗ[A] N.carrier) (hf : Function.Surjective ⇑f) : (M.kernelGrMod N f).ShortExact M N

Any surjection f : M ↠ N of graded modules gives rise to a short exact sequence 0 → ker f → M → N → 0.

@[implicit_reducible]

instance grModCategory {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : CategoryTheory.Category.{u, u + 1} (GrMod 𝒜)

Category instance on graded modules over 𝒜: morphisms are GrMod.Hom, identity and composition come from the underlying linear maps.

noncomputable def tildeProj {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (M : GrMod 𝒜) : (AlgebraicGeometry.Proj 𝒜).Modules

The "tilde" construction on Proj 𝒜: associates to a graded module M a sheaf of O_{Proj 𝒜}-modules M̃.

noncomputable def tildeProj_map {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] {M N : GrMod 𝒜} (f : M.Hom N) : tildeProj 𝒜 M ⟶ tildeProj 𝒜 N

Functoriality of the tilde construction on Proj 𝒜: a graded-module map f induces a morphism of sheaves M̃ → Ñ.

theorem sections_commute_with_filtered_colimits {C : Type (u + 1)} [CategoryTheory.Category.{u, u + 1} C] (X : TopCat) [TopologicalSpace.NoetherianSpace ↑X] {J : Type u} [CategoryTheory.SmallCategory J] [CategoryTheory.IsFiltered J] (F : CategoryTheory.Functor J (TopCat.Sheaf C X)) [CategoryTheory.Limits.HasColimit F] [CategoryTheory.Limits.HasColimitsOfShape J C] (U : TopologicalSpace.Opens ↑X) : have forget := TopCat.Sheaf.forget C X; have sectionsF := F.comp (forget.comp ((CategoryTheory.evaluation (TopologicalSpace.Opens ↑X)ᵒᵖ C).obj (Opposite.op U))); Nonempty ((forget.obj (CategoryTheory.Limits.colimit F)).obj (Opposite.op U) ≅ CategoryTheory.Limits.colimit sectionsF)

On a Noetherian topological space, the sections functor Γ(U, –) commutes with filtered colimits of sheaves: Γ(U, colim Fⱼ) ≅ colim Γ(U, Fⱼ).

theorem tildeProj_exact {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (M₁ M₂ M₃ : GrMod 𝒜) (ses : M₁.ShortExact M₂ M₃) : ∃ (w : CategoryTheory.CategoryStruct.comp (tildeProj_map 𝒜 { toLinearMap := ses.f }) (tildeProj_map 𝒜 { toLinearMap := ses.g }) = 0), { X₁ := tildeProj 𝒜 M₁, X₂ := tildeProj 𝒜 M₂, X₃ := tildeProj 𝒜 M₃, f := tildeProj_map 𝒜 { toLinearMap := ses.f }, g := tildeProj_map 𝒜 { toLinearMap := ses.g }, zero := w }.ShortExact

The tilde functor on Proj 𝒜 preserves short exact sequences of graded modules.

noncomputable def gradedModuleOfSheaf {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (ℱ : (AlgebraicGeometry.Proj 𝒜).Modules) : GrMod 𝒜

Inverse construction: associates to a sheaf ℱ on Proj 𝒜 an underlying graded module (the "Γ_*" graded module of global sections of all twists).

noncomputable def gradedModuleOfSheaf_tildeProj_iso {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (ℱ : (AlgebraicGeometry.Proj 𝒜).Modules) (hqc : IsQuasicoherentSheaf ℱ) : tildeProj 𝒜 (gradedModuleOfSheaf 𝒜 ℱ) ≅ ℱ

Proposition 20 (Lec 14): for a quasi-coherent sheaf ℱ on Proj 𝒜, applying tildeProj to gradedModuleOfSheaf ℱ reproduces ℱ up to isomorphism.

theorem tildeProj_essentiallySurjective {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (ℱ : (AlgebraicGeometry.Proj 𝒜).Modules) (hqc : IsQuasicoherentSheaf ℱ) : ∃ (M : GrMod 𝒜), Nonempty (tildeProj 𝒜 M ≅ ℱ)

Proposition 20 (Lec 14): every quasi-coherent sheaf on Proj 𝒜 is of the form M̃ for some graded module M; equivalently, tildeProj is essentially surjective on quasi-coherent sheaves.

theorem tildeProj_essentiallySurjective_fg {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (ℱ : (AlgebraicGeometry.Proj 𝒜).Modules) (hcoh : IsCoherentSheaf ℱ) : ∃ (M : GrMod 𝒜), M.IsFinitelyGenerated ∧ Nonempty (tildeProj 𝒜 M ≅ ℱ)

Refinement for coherent sheaves: every coherent sheaf on Proj 𝒜 is M̃ for some finitely generated graded module M.

theorem tildeProj_epi_of_graded_surjection {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (M N : GrMod 𝒜) (f : M.carrier →ₗ[A] N.carrier) (hf : Function.Surjective ⇑f) : ∃ (φ : tildeProj 𝒜 M ⟶ tildeProj 𝒜 N), CategoryTheory.Epi φ

A surjection of graded modules M ↠ N induces an epimorphism M̃ ↠ Ñ on Proj 𝒜, obtained from the short exact sequence and right-exactness of tildeProj.

theorem fg_graded_module_quotient_of_shifted_free {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (M : GrMod 𝒜) (hfg : M.IsFinitelyGenerated) : ∃ (d : ℕ) (k : ℕ) (f : ((GrMod.shift 𝒜 (-↑d)).directSumCopies k).carrier →ₗ[A] M.carrier), Function.Surjective ⇑f

Every finitely generated graded module M is a quotient of a finite direct sum of shifted copies of 𝒜, i.e. of 𝒜(-d)^k for some d, k.

theorem coherent_sheaf_quotient_of_twisted_structure_sheaf {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (ℱ : (AlgebraicGeometry.Proj 𝒜).Modules) (hcoh : IsCoherentSheaf ℱ) : ∃ (d : ℕ) (k : ℕ) (φ : tildeProj 𝒜 ((GrMod.shift 𝒜 (-↑d)).directSumCopies k) ⟶ ℱ), CategoryTheory.Epi φ

Corollary 18 (Lec 14) on Proj: every coherent sheaf is a quotient of a direct sum of twists of the structure sheaf, i.e. of O(-d)^k = (𝒜(-d)^k)~.

noncomputable def SerreQuotGrModLocNilp (A : Type u) [CommRing A] (σ : Type u) [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : Type (u + 1)

Serre quotient of the category of graded modules by the Serre subcategory of locally nilpotent graded modules; used to formulate Serre's theorem.

@[implicit_reducible]

noncomputable instance instCategorySerreQuotGrModLocNilp {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : CategoryTheory.Category.{u, u + 1} (SerreQuotGrModLocNilp A σ 𝒜)

Category instance on the Serre quotient GrMod / LocallyNilpotent.

noncomputable def SerreQuotGrModLocNilp.quotientFunctor {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : CategoryTheory.Functor (GrMod 𝒜) (SerreQuotGrModLocNilp A σ 𝒜)

The canonical quotient functor GrMod 𝒜 → GrMod 𝒜 / LocallyNilpotent.

noncomputable def SerreQuotGrModFGFinDim (A : Type u) [CommRing A] (σ : Type u) [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : Type (u + 1)

Serre quotient of finitely generated graded modules by the subcategory of those which are concentrated in finitely many degrees.

@[implicit_reducible]

noncomputable instance instCategorySerreQuotGrModFGFinDim {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : CategoryTheory.Category.{u, u + 1} (SerreQuotGrModFGFinDim A σ 𝒜)

Category instance on the FG Serre quotient.

noncomputable def SerreQuotGrModFGFinDim.quotientFunctor {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : CategoryTheory.Functor (have this := fun (M : GrMod 𝒜) => M.IsFinitelyGenerated; this).FullSubcategory (SerreQuotGrModFGFinDim A σ 𝒜)

Quotient functor on FG graded modules into the corresponding Serre quotient.

@[reducible, inline]

abbrev QCohCat {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : Type (u + 1)

The full subcategory of quasi-coherent sheaves of O_{Proj 𝒜}-modules.

@[reducible, inline]

abbrev CohCat {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : Type (u + 1)

The full subcategory of coherent sheaves of O_{Proj 𝒜}-modules.

theorem tildeProj_zero_iff_locallyNilpotent {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (M : GrMod 𝒜) : CategoryTheory.Limits.IsZero (tildeProj 𝒜 M) ↔ M.IsLocallyNilpotent

A graded module M gives the zero sheaf on Proj 𝒜 iff M is locally nilpotent for the irrelevant ideal.

theorem tildeProj_zero_iff_finiteDimensional {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (M : GrMod 𝒜) (hfg : M.IsFinitelyGenerated) : CategoryTheory.Limits.IsZero (tildeProj 𝒜 M) ↔ M.IsFiniteDimensional

For finitely generated M, M̃ = 0 on Proj 𝒜 iff M is concentrated in finitely many degrees.

theorem tildeProj_faithful_on_quotient {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] {M N : GrMod 𝒜} (f : M.Hom N) (hzero : tildeProj_map 𝒜 f = 0) (x : M.carrier) : ∃ (d : ℕ), ∀ a ∈ (HomogeneousIdeal.irrelevant 𝒜).toIdeal, a ^ d • f.toLinearMap x = 0

Faithfulness modulo the Serre subcategory: a graded-module map whose tilde image is zero on Proj 𝒜 has image annihilated by powers of the irrelevant ideal.

theorem tildeProj_full_on_quotient {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] (M N : GrMod 𝒜) (φ : tildeProj 𝒜 M ⟶ tildeProj 𝒜 N) : ∃ (M' : GrMod 𝒜) (ι : M'.Hom M) (f : M'.Hom N), Function.Injective ⇑ι.toLinearMap ∧ (∀ (x : M.carrier), ∃ (d : ℕ), ∀ a ∈ (HomogeneousIdeal.irrelevant 𝒜).toIdeal, a ^ d • x ∈ ι.toLinearMap.range) ∧ tildeProj_map 𝒜 f = CategoryTheory.CategoryStruct.comp (tildeProj_map 𝒜 ι) φ

Fullness modulo the Serre subcategory: every sheaf morphism M̃ → Ñ on Proj 𝒜 comes from a graded-module morphism out of a "large enough" submodule M' ⊆ M whose inclusion becomes an isomorphism after tildeProj.

noncomputable def serre_equivalence_qcoh {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : SerreQuotGrModLocNilp A σ 𝒜 ≌ QCohCat 𝒜

Serre's theorem (quasi-coherent version): the Serre quotient of graded modules by locally nilpotent modules is equivalent to the category of quasi-coherent sheaves on Proj 𝒜.

noncomputable def serre_equivalence_coh {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : SerreQuotGrModFGFinDim A σ 𝒜 ≌ CohCat 𝒜

Serre's theorem (coherent version): the Serre quotient of finitely generated graded modules by those concentrated in finitely many degrees is equivalent to the category of coherent sheaves on Proj 𝒜.

theorem serre_grothendieck_correspondence {A : Type u} [CommRing A] {σ : Type u} [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : (∀ (ℱ : (AlgebraicGeometry.Proj 𝒜).Modules), IsQuasicoherentSheaf ℱ → ∃ (M : GrMod 𝒜), Nonempty (tildeProj 𝒜 M ≅ ℱ)) ∧ (∀ (M : GrMod 𝒜), CategoryTheory.Limits.IsZero (tildeProj 𝒜 M) ↔ M.IsLocallyNilpotent) ∧ (∀ (M N : GrMod 𝒜) (f : M.Hom N), tildeProj_map 𝒜 f = 0 → ∀ (x : M.carrier), ∃ (d : ℕ), ∀ a ∈ (HomogeneousIdeal.irrelevant 𝒜).toIdeal, a ^ d • f.toLinearMap x = 0) ∧ (∀ (M N : GrMod 𝒜) (φ : tildeProj 𝒜 M ⟶ tildeProj 𝒜 N), ∃ (M' : GrMod 𝒜) (ι : M'.Hom M) (f : M'.Hom N), Function.Injective ⇑ι.toLinearMap ∧ (∀ (x : M.carrier), ∃ (d : ℕ), ∀ a ∈ (HomogeneousIdeal.irrelevant 𝒜).toIdeal, a ^ d • x ∈ ι.toLinearMap.range) ∧ tildeProj_map 𝒜 f = CategoryTheory.CategoryStruct.comp (tildeProj_map 𝒜 ι) φ) ∧ ∀ (M : GrMod 𝒜), M.IsFinitelyGenerated → (CategoryTheory.Limits.IsZero (tildeProj 𝒜 M) ↔ M.IsFiniteDimensional)

Combined Serre–Grothendieck correspondence on Proj 𝒜: essential surjectivity of tildeProj on quasi-coherent sheaves, characterization of the zero objects, faithfulness and fullness up to the Serre subcategory, and the FG/finite-dimensional refinement.