more documentation,test,exports

src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl

@@ -201,8 +201,8 @@ algebraic_cycle(X::AbsCoveredScheme, R::Ring; check::Bool=true) = AlgebraicCycle
 @doc raw"""
     AlgebraicCycle(I::AbsIdealSheaf, R::Ring)
 
-Return the `AlgebraicCycle` ``D = colength(I) ⋅ V(I)`` with coefficients 
-in ``R`` for an equidimensional sheaf of ideals ``I``.
+Return the `AlgebraicCycle` ``D = 1 ⋅ I`` with coefficients
+in ``R`` for a sheaf of equidimensional ideals ``I``.
 """
 function AlgebraicCycle(I::AbsIdealSheaf, R::Ring; check::Bool=true)
   D = AlgebraicCycle(space(I), R; check)
@@ -213,8 +213,11 @@ end
 @doc raw"""
     algebraic_cycle(I::AbsIdealSheaf, R::Ring) -> AlgebraicCycle
 
-Return the `AlgebraicCycle` defined by the equidimensional ideal sheaf `I`. 
+Return the `AlgebraicCycle` ``D = 1 ⋅ I`` with coefficients
+in ``R`` for a sheaf of equidimensional ideals ``I``.
 
+Note that ``I`` must be equidimensional.
+  
 # Examples
 ```jldoctest
 julia> P, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);
@@ -236,16 +239,16 @@ given as the formal sum of
 
 ```
 """
-algebraic_cycle(I::AbsIdealSheaf, R::Ring) = AlgebraicCycle(I, R)
+algebraic_cycle(I::AbsIdealSheaf, R::Ring; check::Bool=true) = AlgebraicCycle(I, R; check)
 
 @doc raw"""
     AlgebraicCycle(I::AbsIdealSheaf)
 
 Return the `AlgebraicCycle` ``D = 1 ⋅ I`` with coefficients
 in ``ℤ`` for a sheaf of equidimensional ideals ``I``.
 """
-function AlgebraicCycle(I::AbsIdealSheaf)
-  D = AlgebraicCycle(space(I), ZZ)
+function AlgebraicCycle(I::AbsIdealSheaf; check::Bool=true)
+  D = AlgebraicCycle(space(I), ZZ; check)
   D[I] = one(ZZ)
   return D
 end
@@ -276,7 +279,7 @@ given as the formal sum of
   1 * sheaf of ideals
 ```
 """
-algebraic_cycle(I::AbsIdealSheaf) = AlgebraicCycle(I)
+algebraic_cycle(I::AbsIdealSheaf; check::Bool=true) = AlgebraicCycle(I; check)
 
 ### copy constructor
 function copy(D::AlgebraicCycle) 

src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl

@@ -268,12 +268,15 @@ end
 ### (specialized variant of associated_points, using pure codimension 1
 ###  and taking multiplicities into account)
 @doc raw"""
-    irreducible_decomposition(C::EffectiveCartierDivisor)
+    irreducible_decomposition(C::EffectiveCartierDivisor; check::Bool=true) -> AbsWeilDivisor
 
-Return a `Vector` of pairs ``(I,k)`` corresponding to the irreducible components of ``C``. More precisely,  each ``I`` is a prime  `AbsIdealSheaf` corresponding to an irreducible component of ``C`` and ``k``is the multiplicity of this component in ``C``.
+Return $C$ as a linear combination of prime Weil divisors.
+
+Assumes that the ambient scheme is integral and locally noetherian.
 """
-function irreducible_decomposition(C::EffectiveCartierDivisor)
+function irreducible_decomposition(C::EffectiveCartierDivisor; check::Bool=true)
   X = ambient_scheme(C)
+  @check is_integral(X) "ambient scheme must be integral"
   cov = default_covering(X)
   OOX = OO(X)
 
@@ -301,7 +304,8 @@ function irreducible_decomposition(C::EffectiveCartierDivisor)
       push!(associated_primes_temp, (I_sheaf_temp, saturation_index))
     end
   end
-  return(associated_primes_temp)
+  D = WeilDivisor(X, ZZ, IdDict(associated_primes_temp); check=false)
+  return D
 end
 
 ### Conversion into WeilDivisors

src/AlgebraicGeometry/Schemes/Divisors/Types.jl

@@ -31,7 +31,7 @@
       #is_separated(X) || error("scheme must be separated") # We need to test this somehow, but how?
       d = isempty(coefficients) ? 0 : dim(first(keys(coefficients)))
       for D in keys(coefficients)
-        (is_equidimensional(D) && dim(D) == d) || error("components of a cycle must be sheaves of equidimensional ideals")
+        (is_equidimensional(D) && dim(D) == d) || error("components of a cycle must be sheaves of equidimensional ideals of the same dimension")
       end
       true
     end
@@ -46,7 +46,7 @@ end
 @doc raw"""
     WeilDivisor <: AbsWeilDivisor
 
-A Weil divisor on an integral separated `AbsCoveredScheme` ``X``; 
+A Weil divisor on an integral `AbsCoveredScheme` ``X``; 
 stored as a formal linear combination over some ring ``R`` of 
 ideal sheaves on ``X``.
 """
@@ -114,10 +114,9 @@ end
 @doc raw""" 
     EffectiveCartierDivisor{CoveredSchemeType<:AbsCoveredScheme}
 
-An effective Cartier divisor on a scheme $X$ is a closed subscheme $D \subseteq S$ whose ideal sheaf $\mathcal{I}_D \subseteq \mathcal{O}_S$ is an invertible $\mathcal{O}_S$-module. In particular, $\mathcal{I}_D$ is locally principal.
+An effective Cartier divisor on a scheme $X$ is a closed subscheme $D \subseteq X$ whose ideal sheaf $\mathcal{I}_D \subseteq \mathcal{O}_X$ is an invertible $\mathcal{O}_X$-module. In particular, $\mathcal{I}_D$ is locally principal.
 
-
-See [`trivializing_covering(C::EffectiveCartierDivisor)`](@ref) for how to
+Internally, $C$ stores a [`trivializing_covering(C::EffectiveCartierDivisor)`](@ref).
 The scheme $X$ is of type `CoveredSchemeType`.
 """
 @attributes mutable struct EffectiveCartierDivisor{

src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl

@@ -71,10 +71,10 @@ coefficients in the integer ring.
 ```jldoctest
 julia> P, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);
 
-julia> I = ideal([x^3-y^2*z]);
-
 julia> Y = proj(P);
 
+julia> I = ideal([(x^3-y^2*z)]);
+
 julia> II = IdealSheaf(Y, I);
 
 julia> weil_divisor(II)
@@ -83,6 +83,23 @@ Effective weil divisor
 with coefficients in integer ring
 given as the formal sum of
   1 * sheaf of ideals
+  
+julia> JJ = II^2;
+
+julia> D = weil_divisor(JJ)
+Effective weil divisor
+  on scheme over QQ covered with 3 patches
+with coefficients in integer ring
+given as the formal sum of
+  1 * product of 2 ideal sheaves
+
+julia> irreducible_decomposition(D)
+Effective weil divisor
+  on scheme over QQ covered with 3 patches
+with coefficients in integer ring
+given as the formal sum of
+  2 * sheaf of prime ideals
+
 ```
 """
 weil_divisor(I::AbsIdealSheaf; check::Bool=true) = WeilDivisor(I, check=check)

src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl

@@ -1031,6 +1031,29 @@ function maximal_associated_points(
   return comps
 end
 
+@doc raw"""
+    minimal_primes(I::AbsIdealSheaf; kwargs...)
+
+Return the minimal prime ideal sheaves of ``I``.
+
+These are the ideal sheaves of the maximal associated points of ``I``.
+
+See [`maximal_associated_points`](@ref) for possible keyword arguments.
+"""
+minimal_primes(I::AbsIdealSheaf; kwargs...) = maximal_associated_points(I; kwargs...)
+
+@doc raw"""
+    associated_primes(I::AbsIdealSheaf; kwargs...)
+
+Return the prime ideal sheaves associated to``I``.
+
+These are the ideal sheaves of the associated points of ``I``.
+
+See [`associated_points`](@ref) for possible keyword arguments.
+"""
+associated_primes(I::AbsIdealSheaf; kwargs...) = associated_points(I; kwargs...)
+
+
 @doc raw"""
     associated_points(I::AbsIdealSheaf)
 
@@ -1283,40 +1306,26 @@ function Base.show(io::IO, ::MIME"text/plain", I::SumIdealSheaf)
   io = pretty(io)
   print(io, "Sum of \n")
   print(io, Indent())
-  for J in summands(I)
-    print(io, J, "\n")
-  end
+  join(io, summands(I), "\n")
   print(io, Dedent())
 end
 
 function Base.show(io::IO, ::MIME"text/plain", I::ProductIdealSheaf)
   io = pretty(io)
   print(io, "Product of \n")
   print(io, Indent())
-  for J in factors(I)
-    print(io, J, "\n")
-  end
+  join(io, factors(I), "\n")
   print(io, Dedent())
 end
 
 function Base.show(io::IO, I::SumIdealSheaf)
   io = pretty(io)
-  print(io, "Sum of \n")
-  print(io, Indent())
-  for J in summands(I)
-    print(io, J, "\n")
-  end
-  print(io, Dedent())
+  print(io, "Sum of $(length(summands(I))) ideal sheaves")
 end
 
 function Base.show(io::IO, I::ProductIdealSheaf)
   io = pretty(io)
-  print(io, "Product of \n")
-  print(io, Indent())
-  for J in factors(I)
-    print(io, J, "\n")
-  end
-  print(io, Dedent())
+  print(io, "Product of $(length(factors(I))) ideal sheaves")
 end
 
 function Base.show(io::IO, I::PrimeIdealSheafFromChart)

src/exports.jl

@@ -276,6 +276,8 @@ export as_dictionary
 export as_gset
 export ascending_compositions
 export associahedron
+export associated_primes
+export associated_points
 export atlas_description
 export atlas_group
 export atlas_irrationality
@@ -756,6 +758,7 @@ export invert_birational_map
 export inverted_set
 export invlex
 export irreducible_components
+export irreducible_decomposition
 export irreducible_secondary_invariants
 export irreducibles
 export irrelevant_ideal
@@ -1033,6 +1036,7 @@ export matroid_hex
 export max_GC_rank_polytope
 export maxes
 export maximal_abelian_quotient, has_maximal_abelian_quotient, set_maximal_abelian_quotient
+export maximal_associated_points
 export maximal_blocks
 export maximal_cells
 export maximal_cones

src/forward_declarations.jl

@@ -38,6 +38,37 @@ For an equidimensional ideal sheaf $\mathcal{I}$ its interpretation as a cycle i
 Let $V(\mathcal{I})=E_{1} \cup \dots E_{n}$ be the decomposition of the vanishing locus
 of $\mathcal{I}$ into irreducible components $E_i=V(\mathcal{P}_i)$ with $\mathcal{P}_i$ prime. 
 Then $E$ corresponds to the cycle $D = \sum_{i=1}^{n} \mathrm{colength}_{\mathcal{P}_i}(\mathcal{I})E_i$.
+
+# Examples
+```jldoctest
+julia> P2 = projective_space(QQ,2); (s0,s1,s2) = homogeneous_coordinates(P2);
+
+julia> I = ideal_sheaf(P2,ideal([s0,s1^2]))
+Sheaf of ideals
+  on scheme over QQ covered with 3 patches
+    1: [(s1//s0), (s2//s0)]   affine 2-space
+    2: [(s0//s1), (s2//s1)]   affine 2-space
+    3: [(s0//s2), (s1//s2)]   affine 2-space
+with restrictions
+  1: Ideal (1, (s1//s0)^2)
+  2: Ideal ((s0//s1), 1)
+  3: Ideal ((s0//s2), (s1//s2)^2)
+
+julia> D = algebraic_cycle(I)
+Effective algebraic cycle
+  on scheme over QQ covered with 3 patches
+with coefficients in integer ring
+given as the formal sum of
+  1 * sheaf of ideals
+
+julia> irreducible_decomposition(D)
+Effective algebraic cycle
+  on scheme over QQ covered with 3 patches
+with coefficients in integer ring
+given as the formal sum of
+  2 * sheaf of prime ideals
+
+```
 """
 abstract type AbsAlgebraicCycle{
                                 CoveredSchemeType<:AbsCoveredScheme, 
@@ -49,6 +80,41 @@ end
     AbsWeilDivisor{CoveredSchemeType, CoefficientRingType} <: AbsAlgebraicCycle{CoveredSchemeType, CoefficientRingType}
 
 A Weil divisor with coefficients of type `CoefficientRingType` on a (locally) Noetherian integral scheme ``X``  of type `CoveredSchemeType`.
+
+# Examples
+```jldoctest
+julia> P2 = projective_space(QQ,2); (s0,s1,s2) = homogeneous_coordinates(P2);
+
+julia> I = ideal((s0*s1)^2);
+
+julia> II = ideal_sheaf(P2, I);
+
+julia> D = weil_divisor(II)
+Effective weil divisor
+  on scheme over QQ covered with 3 patches
+with coefficients in integer ring
+given as the formal sum of
+  1 * sheaf of ideals
+
+julia> E = irreducible_decomposition(D)
+Effective weil divisor
+  on scheme over QQ covered with 3 patches
+with coefficients in integer ring
+given as the formal sum of
+  2 * prime ideal sheaf on scheme over QQ covered with 3 patches extended from ideal ((s1//s0)) on affine 2-space
+  2 * prime ideal sheaf on scheme over QQ covered with 3 patches extended from ideal ((s0//s1)) on affine 2-space
+
+julia> P = components(E)[1]
+Prime ideal sheaf on Scheme over QQ covered with 3 patches extended from Ideal ((s1//s0)) on Affine 2-space
+
+julia> components(D)[1] == II
+true
+
+julia> D[II] # to get the coefficient
+1
+
+julia> P[I]
+```
 """
 abstract type AbsWeilDivisor{CoveredSchemeType, CoefficientRingType} <: AbsAlgebraicCycle{CoveredSchemeType, CoefficientRingType} end
 

test/AlgebraicGeometry/Schemes/WeilDivisor.jl

@@ -123,11 +123,10 @@ end
   KK = function_field(X)
   R = ambient_coordinate_ring(representative_patch(KK))
   x = gens(R)
-  @test in_linear_system(KK(x[1]), D)
-  @test !in_linear_system(KK(x[1]^2), D)
-  @test in_linear_system(KK(x[1]^2), 2*D)
-  # Not running at the moment; work in progress
-  #@test !in_linear_system(KK(x[1], x[2]), D)
+  @test is_in_linear_system(KK(x[1]), D)
+  @test !is_in_linear_system(KK(x[1]^2), D)
+  @test is_in_linear_system(KK(x[1]^2), 2*D)
+  @test !is_in_linear_system(KK(x[1], x[2]), D)
 
   L = LinearSystem(KK.([1, x[1], x[2], x[1]^2, x[1]*x[2], x[2]^2]), 2*D)
   H = S[1]+S[2]+S[3]
@@ -160,7 +159,7 @@ end
   U = X[1][2]
   u, v = gens(OO(U))
   f = KK(u, v)
-  @test !in_linear_system(f, D)
+  @test !is_in_linear_system(f, D)
 end
 
 @testset "intersections of weil divisors on surfaces" begin