FormalConjecturesForMathlib.AlgebraicGeometry.VectorBundle

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module

public import Mathlib.Algebra.Category.ModuleCat.Sheaf.Free
public import Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous
public import Mathlib.CategoryTheory.Sites.CoversTop.Basic

@[expose] public sectionuniverse u vnamespace SheafOfModulesopen CategoryTheory Limitsvariable {C : Type u} [Category.{v, u} C]variable {J : GrothendieckTopology C}variable {R : Sheaf J RingCat} (M : SheafOfModules R)variable [∀ (X : C), (J.over X).HasSheafCompose (forget₂ RingCat AddCommGrpCat)]variable [∀ (X : C), HasWeakSheafify (J.over X) AddCommGrpCat]variable [∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat]

A vector bundle is a sheaf of modules that is locally isomorphic to a free sheaf.

structure VectorBundleData (M : SheafOfModules.{v} R) where

The index type for the open cover on which the vector bundle is free.

OpensIndex : Type u

The indexed family of opens on which the vector bundle is free.

opens : OpensIndex → C

The indexed family of opens covers the base.

coversTop : J.CoversTop opens

The indexed family of types that index local bases for the vector bundle.

BasisIndex : OpensIndex → Type v

The restriction of the vector bundle to one of the opens is free.

locallyFree : ∀ i, (M.over <| opens i) ≅ SheafOfModules.free (BasisIndex i)structure FiniteRankVectorBundleData (M : SheafOfModules R) extends VectorBundleData M where

The predicate that every local basis in the defining data of the vector bundle is finite.

finite : ∀ i, Finite (BasisIndex i)

A class for vector bundles of constant finite rank.

class IsVectorBundleWithRank (M : SheafOfModules R) (n : ℕ) where

The predicate that a sheaf of modules is locally free.

  exists_vector_bundle_data : ∃ (D : M.FiniteRankVectorBundleData),
    ∀ i, Nat.card (D.BasisIndex i) = n

end SheafOfModules