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Some more documentation for coherent sheaves (#4352)
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* Add more documentation for sheaves.
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@HechtiDerLachs
HechtiDerLachs authored Nov 27, 2024
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21 changes: 21 additions & 0 deletions docs/src/AlgebraicGeometry/Schemes/Sheaves.md
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Expand Up @@ -22,15 +22,36 @@ StructureSheafOfRings
AbsIdealSheaf
IdealSheaf
PrimeIdealSheafFromChart
SumIdealSheaf
ProductIdealSheaf
SimplifiedIdealSheaf
PullbackIdealSheaf
RadicalOfIdealSheaf
ToricIdealSheafFromCoxRingIdeal
SingularLocusIdealSheaf
```

## Coherent sheaves of modules

These are some types for coherent sheaves.
```@docs
SheafOfModules
HomSheaf
PushforwardSheaf
PullbackSheaf
DirectSumSheaf
```

We provide some common constructors.
```@docs
twisting_sheaf(IP::AbsProjectiveScheme{<:Field}, d::Int)
tautological_bundle(IP::AbsProjectiveScheme{<:Field})
cotangent_sheaf(X::AbsCoveredScheme)
free_module(R::StructureSheafOfRings, n::Int)
dual(M::SheafOfModules)
tangent_sheaf(X::AbsCoveredScheme)
```

```@docs
projectivization(E::AbsCoherentSheaf; var_names::Vector{String}=Vector{String}(), check::Bool=true)
```
277 changes: 277 additions & 0 deletions src/AlgebraicGeometry/Schemes/Sheaves/CoherentSheaves.jl
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Expand Up @@ -341,6 +341,37 @@ end
For an `AbsCoveredScheme` ``X``, return the sheaf ``Ω¹(X)`` of Kaehler-differentials
on ``X`` as a `CoherentSheaf`.
# Examples
```jldoctest
julia> IP1 = projective_space(QQ, 1);
julia> X = covered_scheme(IP1)
Scheme
over rational field
with default covering
described by patches
1: affine 1-space
2: affine 1-space
in the coordinate(s)
1: [(s1//s0)]
2: [(s0//s1)]
julia> Omega = cotangent_sheaf(X);
julia> U, V = affine_charts(X);
julia> UV, VU = gluing_domains(default_covering(X)[U, V]);
julia> dx = Omega(U)[1]
d(s1//s0)
julia> Omega(V)
Free module of rank 1 over multivariate polynomial ring in 1 variable over QQ
julia> Omega(U, VU)(dx)
-1/(s0//s1)^2*d(s0//s1)
```
"""
@attr SheafOfModules function cotangent_sheaf(X::AbsCoveredScheme)
MD = IdDict{AbsAffineScheme, ModuleFP}()
Expand Down Expand Up @@ -419,6 +450,53 @@ end
in the constructor.
=#

@doc raw"""
HomSheaf
For two `AbsCoherentSheaf`s `F` and `G` on an `AbsCoveredScheme` `X`
this computes the sheaf associated to `U -> Hom(F(U), G(U))`.
# Examples
```jldoctest
julia> IP1 = projective_space(GF(7), [:x, :y])
Projective space of dimension 1
over prime field of characteristic 7
with homogeneous coordinates [x, y]
julia> Y = covered_scheme(IP1);
julia> Omega = cotangent_sheaf(Y)
Coherent sheaf of modules
on scheme over GF(7) covered with 2 patches
1: [(y//x)] affine 1-space
2: [(x//y)] affine 1-space
with restrictions
1: free module of rank 1 over multivariate polynomial ring in 1 variable over GF(7)
2: free module of rank 1 over multivariate polynomial ring in 1 variable over GF(7)
julia> F = free_module(OO(Y), 1)
Coherent sheaf of modules
on scheme over GF(7) covered with 2 patches
1: [(y//x)] affine 1-space
2: [(x//y)] affine 1-space
with restrictions
1: free module of rank 1 over multivariate polynomial ring in 1 variable over GF(7)
2: free module of rank 1 over multivariate polynomial ring in 1 variable over GF(7)
julia> T = Oscar.HomSheaf(Omega, F)
Coherent sheaf of modules
on scheme over GF(7) covered with 2 patches
1: [(y//x)] affine 1-space
2: [(x//y)] affine 1-space
with restrictions
1: hom of (Multivariate polynomial ring in 1 variable over GF(7)^1, Multivariate polynomial ring in 1 variable over GF(7)^1)
2: hom of (Multivariate polynomial ring in 1 variable over GF(7)^1, Multivariate polynomial ring in 1 variable over GF(7)^1)
julia> typeof(T)
Oscar.HomSheaf{CoveredScheme{FqField}, AbsAffineScheme, ModuleFP, Map}
```
"""
@attributes mutable struct HomSheaf{SpaceType, OpenType, OutputType,
RestrictionType
} <: AbsCoherentSheaf{
Expand Down Expand Up @@ -457,6 +535,53 @@ codomain(M::HomSheaf) = M.codomain
########################################################################
# Sheaves of direct sums #
########################################################################
@doc raw"""
DirectSumSheaf
Given two or more `AbsCoherentSheaf`s `F` and `G` on an `AbsCoveredScheme` `X`,
this holds the sheaf associated to the direct sum of `F` and `G`
# Examples
```jldoctest
julia> IP1 = projective_space(GF(7), [:x, :y])
Projective space of dimension 1
over prime field of characteristic 7
with homogeneous coordinates [x, y]
julia> Y = covered_scheme(IP1);
julia> Omega = cotangent_sheaf(Y)
Coherent sheaf of modules
on scheme over GF(7) covered with 2 patches
1: [(y//x)] affine 1-space
2: [(x//y)] affine 1-space
with restrictions
1: free module of rank 1 over multivariate polynomial ring in 1 variable over GF(7)
2: free module of rank 1 over multivariate polynomial ring in 1 variable over GF(7)
julia> F = free_module(OO(Y), 1)
Coherent sheaf of modules
on scheme over GF(7) covered with 2 patches
1: [(y//x)] affine 1-space
2: [(x//y)] affine 1-space
with restrictions
1: free module of rank 1 over multivariate polynomial ring in 1 variable over GF(7)
2: free module of rank 1 over multivariate polynomial ring in 1 variable over GF(7)
julia> W = Oscar.DirectSumSheaf(Y, [Omega, F])
Coherent sheaf of modules
on scheme over GF(7) covered with 2 patches
1: [(y//x)] affine 1-space
2: [(x//y)] affine 1-space
with restrictions
1: direct sum of (Multivariate polynomial ring in 1 variable over GF(7)^1, Multivariate polynomial ring in 1 variable over GF(7)^1)
2: direct sum of (Multivariate polynomial ring in 1 variable over GF(7)^1, Multivariate polynomial ring in 1 variable over GF(7)^1)
julia> typeof(W)
DirectSumSheaf{CoveredScheme{FqField}, AbsAffineScheme, ModuleFP, Map}
```
"""
@attributes mutable struct DirectSumSheaf{SpaceType, OpenType, OutputType,
RestrictionType
} <: AbsCoherentSheaf{
Expand Down Expand Up @@ -587,6 +712,91 @@ end
# It is clear that this can and should be made lazy.
# =#

@doc raw"""
PushforwardSheaf
For a `CoveredClosedEmbedding` `i : X -> Y` and an `AbsCoherentSheaf` `F`
on `X` this computes the coherent sheaf `i_* F` on `Y`.
# Examples
```jldoctest
julia> IP2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> S = cox_ring(IP2)
Multivariate polynomial ring in 3 variables over QQ graded by
x1 -> [1]
x2 -> [1]
x3 -> [1]
julia> x, y, z = gens(S);
julia> I = ideal(S, x^3 - y*z^2);
julia> II = ideal_sheaf(IP2, I);
julia> X, inc_X = sub(II);
julia> F = cotangent_sheaf(X)
Coherent sheaf of modules
on scheme over QQ covered with 3 patches
1: [x_1_1, x_2_1] scheme(x_1_1^3 - x_2_1)
2: [x_1_2, x_2_2] scheme(x_1_2^2*x_2_2 - 1)
3: [x_1_3, x_2_3] scheme(x_1_3^3 - x_2_3^2)
with restrictions
1: subquotient of submodule with 2 generators
1: dx_1_1
2: dx_2_1
by submodule with 1 generator
1: 3*x_1_1^2*dx_1_1 - dx_2_1
2: subquotient of submodule with 2 generators
1: dx_1_2
2: dx_2_2
by submodule with 1 generator
1: 2*x_1_2*x_2_2*dx_1_2 + x_1_2^2*dx_2_2
3: subquotient of submodule with 2 generators
1: dx_1_3
2: dx_2_3
by submodule with 1 generator
1: 3*x_1_3^2*dx_1_3 - 2*x_2_3*dx_2_3
julia> inc_F = Oscar.PushforwardSheaf(inc_X, F)
Coherent sheaf of modules
on normal toric variety
with restrictions
1: subquotient of submodule with 2 generators
1: dx_1_1
2: dx_2_1
by submodule with 5 generators
1: (x_1_1^3 - x_2_1)*dx_1_1
2: (x_1_1^3 - x_2_1)*dx_2_1
3: 3*x_1_1^2*dx_1_1 - dx_2_1
4: (x_1_1^3 - x_2_1)*dx_1_1
5: (x_1_1^3 - x_2_1)*dx_2_1
2: subquotient of submodule with 2 generators
1: dx_1_2
2: dx_2_2
by submodule with 5 generators
1: (x_1_2^2*x_2_2 - 1)*dx_1_2
2: (x_1_2^2*x_2_2 - 1)*dx_2_2
3: 2*x_1_2*x_2_2*dx_1_2 + x_1_2^2*dx_2_2
4: (x_1_2^2*x_2_2 - 1)*dx_1_2
5: (x_1_2^2*x_2_2 - 1)*dx_2_2
3: subquotient of submodule with 2 generators
1: dx_1_3
2: dx_2_3
by submodule with 5 generators
1: (x_1_3^3 - x_2_3^2)*dx_1_3
2: (x_1_3^3 - x_2_3^2)*dx_2_3
3: 3*x_1_3^2*dx_1_3 - 2*x_2_3*dx_2_3
4: (x_1_3^3 - x_2_3^2)*dx_1_3
5: (x_1_3^3 - x_2_3^2)*dx_2_3
julia> typeof(inc_F)
PushforwardSheaf{NormalToricVariety, AbsAffineScheme, ModuleFP, Map}
```
"""
@attributes mutable struct PushforwardSheaf{SpaceType, OpenType, OutputType,
RestrictionType
} <: AbsCoherentSheaf{
Expand Down Expand Up @@ -670,6 +880,73 @@ end
# to create a module for ℳ (U) when U ⊂ X is an `affine_chart` of X.
# The user is hence forced to work in the refinement only.


@doc raw"""
PullbackSheaf
For a morphism `f : X -> Y` of `AbsCoveredScheme`s and a coherent
sheaf `F` on `Y` this computes the pullback `f^* F` on `X`.
# Examples
```jldoctest
julia> IP2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> S = cox_ring(IP2)
Multivariate polynomial ring in 3 variables over QQ graded by
x1 -> [1]
x2 -> [1]
x3 -> [1]
julia> x, y, z = gens(S);
julia> I = ideal(S, x^3 - y*z^2);
julia> II = ideal_sheaf(IP2, I);
julia> X, inc_X = sub(II);
julia> F = cotangent_sheaf(codomain(inc_X))
Coherent sheaf of modules
on normal toric variety
with restrictions
1: submodule with 2 generators
1: dx_1_1
2: dx_2_1
represented as subquotient with no relations
2: submodule with 2 generators
1: dx_1_2
2: dx_2_2
represented as subquotient with no relations
3: submodule with 2 generators
1: dx_1_3
2: dx_2_3
represented as subquotient with no relations
julia> inc_F = Oscar.PullbackSheaf(inc_X, F)
Coherent sheaf of modules
on scheme over QQ covered with 3 patches
1: [x_1_1, x_2_1] scheme(x_1_1^3 - x_2_1)
2: [x_1_2, x_2_2] scheme(x_1_2^2*x_2_2 - 1)
3: [x_1_3, x_2_3] scheme(x_1_3^3 - x_2_3^2)
with restrictions
1: submodule with 2 generators
1: dx_1_1
2: dx_2_1
represented as subquotient with no relations
2: submodule with 2 generators
1: dx_1_2
2: dx_2_2
represented as subquotient with no relations
3: submodule with 2 generators
1: dx_1_3
2: dx_2_3
represented as subquotient with no relations
julia> typeof(inc_F)
PullbackSheaf{CoveredScheme{QQField}, AbsAffineScheme, ModuleFP, Map}
```
"""
@attributes mutable struct PullbackSheaf{SpaceType, OpenType, OutputType,
RestrictionType
} <: AbsCoherentSheaf{
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