HartshorneConjecture.lean`

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9 declarations, 2 contain sorry, 0 declared as axioms

Hartshorne's conjecture on Vector Bundles

References:

[Har1974] R. Hartshorne, Varieties of small codimension in projective space.
[MO2010] Evidences on Hartshorne's conjecture? References?

Imports (1)

FormalConjecturesUtil

structure AlgebraicGeometry.Scheme.VectorBundles A vector bundle over a scheme S is a locally free $\mathcal{O}_S$-module of finite rank line 39 source
Source documentation
A vector bundle over a scheme S is a locally free $\mathcal{O}_S$-module of finite rank.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AlgebraicGeometry.Scheme
```
structure VectorBundles where
…
```
instance AlgebraicGeometry.Scheme.(instance at line 45) line 45 source
instance AlgebraicGeometry.Scheme.(instance at line 51) line 51 source
Source documentation
Vector bundles form a category.
```
instance : Category S.VectorBundles :=
…
```
def AlgebraicGeometry.Scheme.VectorBundles.toModule line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AlgebraicGeometry.Scheme, CategoryTheory.Functor, AlgebraicGeometry.Scheme.VectorBundles, AlgebraicGeometry.Scheme.Modules
Proof uses: CategoryTheory.Functor.mk, AlgebraicGeometry.Scheme.VectorBundles.carrier, Quiver.Hom, CategoryTheory.InducedCategory.Hom.hom
8 instances, coercions and auxiliary constants left out.
theorem AlgebraicGeometry.Scheme.hasFiniteCoproductsVectorBundles sorry line 59 source
depends on sorry what this means
Statement uses: AlgebraicGeometry.Scheme, CategoryTheory.Limits.HasFiniteCoproducts, AlgebraicGeometry.Scheme.VectorBundles
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
instance AlgebraicGeometry.Scheme.(instance at line 62) line 62 source
structure AlgebraicGeometry.Scheme.VectorBundles.Splitting A splitting of a vector bundle 𝓕 is a non-trivial direct sum decomposition of 𝓕 line 69 source
Source documentation
A splitting of a vector bundle 𝓕 is a non-trivial direct sum decomposition of 𝓕

depends on sorry what this means
Statement uses: AlgebraicGeometry.Scheme, AlgebraicGeometry.Scheme.VectorBundles, Fintype, Nonempty
```
structure VectorBundles.Splitting (𝓕 : S.VectorBundles) (ι : Type) [Fintype ι] [Nonempty ι] where
…
```
instance AlgebraicGeometry.Scheme.(instance at line 74) line 74 source
theorem AlgebraicGeometry.Scheme.harthshorne_conjecture There are no indecomposable vector bundles of rank 2 on $\mathbb{P}^n$ for $n \ge 7$ sorry line 85 source
Source documentation
There are no indecomposable vector bundles of rank 2 on $\mathbb{P}^n$ for $n \ge 7$. This is Conjecture 6.3 in [Har1974].

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, AlgebraicGeometry.Scheme.VectorBundles, ProjectiveSpace, Fin, HAdd.hAdd, AlgebraicGeometry.Spec, CommRingCat.of, Complex, Eq, AlgebraicGeometry.Scheme.VectorBundles.rank, Nonempty, AlgebraicGeometry.Scheme.VectorBundles.Splitting
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem harthshorne_conjecture (n : ℕ) (hn : 7 ≤ n)
    (𝓕 : VectorBundles ℙ(Fin (n + 1); Spec (.of ℂ)))
    (h𝓕 : 𝓕.rank = 2) :
    Nonempty (𝓕.Splitting (Fin 2)) := by
…
```
Mentions: AlgebraicGeometry.Scheme.VectorBundles