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Sb/print local rings #2806

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Sep 17, 2023
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Sb/print local rings #2806

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e45674e
update printing for local rings and schemes
simonbrandhorst Sep 15, 2023
f549b08
fix doctest
simonbrandhorst Sep 16, 2023
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15 changes: 8 additions & 7 deletions docs/src/CommutativeAlgebra/affine_algebras.md
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Expand Up @@ -228,7 +228,8 @@ ideal(x - y, z^4)

julia> base_ring(a)
Quotient
of multivariate polynomial ring in 3 variables over QQ
of multivariate polynomial ring in 3 variables x, y, z
over rational field
by ideal(-x^2 + y, -x^3 + z)

julia> gens(a)
Expand Down Expand Up @@ -378,7 +379,7 @@ julia> V2 = [p2(w-y), p2(x), p2(z)];
julia> proj = hom(D2, C2, V2)
Ring homomorphism
from graded multivariate polynomial ring in 3 variables over QQ
to quotient of multivariate polynomial ring by ideal with 3 generators
to quotient of multivariate polynomial ring by ideal(-x*z + y^2, -w*z + x*y, -w*y + x^2)
defined by
a -> w - y
b -> x
Expand Down Expand Up @@ -437,7 +438,7 @@ julia> V = [p(2*a + b^6), p(7*b - a^2), p(c^2)];
julia> F = hom(D, C, V)
Ring homomorphism
from multivariate polynomial ring in 3 variables over QQ
to quotient of multivariate polynomial ring by ideal with 1 generator
to quotient of multivariate polynomial ring by ideal(-b^3 + c)
defined by
x -> 2*a + c^2
y -> -a^2 + 7*b
Expand Down Expand Up @@ -519,17 +520,17 @@ julia> L[1]

julia> L[2]
Ring homomorphism
from quotient of multivariate polynomial ring by ideal with 2 generators
to quotient of multivariate polynomial ring by ideal with 2 generators
from quotient of multivariate polynomial ring by ideal(x*y, x*z)
to quotient of multivariate polynomial ring by ideal(2*x^2 + x*y, 10*x^2 + 5*x*y + x*z)
defined by
x -> x
y -> 2*x + y
z -> 10*x + 5*y + z

julia> L[3]
Ring homomorphism
from quotient of multivariate polynomial ring by ideal with 2 generators
to quotient of multivariate polynomial ring by ideal with 2 generators
from quotient of multivariate polynomial ring by ideal(2*x^2 + x*y, 10*x^2 + 5*x*y + x*z)
to quotient of multivariate polynomial ring by ideal(x*y, x*z)
defined by
x -> x
y -> -2*x + y
Expand Down
8 changes: 6 additions & 2 deletions docs/src/CommutativeAlgebra/localizations.md
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Expand Up @@ -303,7 +303,8 @@ x/y

julia> parent(f)
Localization
of multivariate polynomial ring in 3 variables over QQ
of multivariate polynomial ring in 3 variables x, y, z
over rational field
at complement of prime ideal(x)

julia> g = iota(y)/iota(z)
Expand Down Expand Up @@ -348,7 +349,10 @@ x

julia> parent(f)
Localization
of quotient of multivariate polynomial ring by ideal with 2 generators
of quotient
of multivariate polynomial ring in 2 variables x, y
over number field of degree 2 over QQ
by ideal(2*x^2 - y^3, 2*x^2 - y^5)
at complement of prime ideal(y - 1, x - a)

julia> g = f/phi(y)
Expand Down
4 changes: 2 additions & 2 deletions experimental/PlaneCurve/src/PlaneCurve.jl
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Expand Up @@ -469,8 +469,8 @@ julia> C = AffinePlaneCurve(y^2+x-x^3)
Affine plane curve defined by -x^3 + x + y^2

julia> Oscar.ring(C)
(Quotient of multivariate polynomial ring by ideal with 1 generator, Map from
Multivariate polynomial ring in 2 variables over QQ to Quotient of multivariate polynomial ring by ideal with 1 generator defined by a julia-function with inverse)
(Quotient of multivariate polynomial ring by ideal(-x^3 + x + y^2), Map from
Multivariate polynomial ring in 2 variables over QQ to Quotient of multivariate polynomial ring by ideal(-x^3 + x + y^2) defined by a julia-function with inverse)
```
"""
function ring(C::PlaneCurve)
Expand Down
16 changes: 8 additions & 8 deletions experimental/Schemes/BlowupMorphism.jl
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Expand Up @@ -37,14 +37,14 @@ ideal(x, y, z)
julia> bl = blow_up(A3, I)
Blow up
of scheme over QQ covered with 1 patch
1b: [x, y, z] spec of multivariate polynomial ring
1b: [x, y, z] affine 3-space
in sheaf of ideals with restriction
1b: ideal(x, y, z)
with domain
scheme over QQ covered with 3 patches
1a: [(s1//s0), (s2//s0), x] spec of quotient of multivariate polynomial ring
2a: [(s0//s1), (s2//s1), y] spec of quotient of multivariate polynomial ring
3a: [(s0//s2), (s1//s2), z] spec of quotient of multivariate polynomial ring
1a: [(s1//s0), (s2//s0), x] V(0, 0, 0)
2a: [(s0//s1), (s2//s1), y] V(0, 0, 0)
3a: [(s0//s2), (s1//s2), z] V(0, 0, 0)
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and exceptional divisor
effective cartier divisor defined by
sheaf of ideals with restrictions
Expand All @@ -55,9 +55,9 @@ and exceptional divisor
julia> E = exceptional_divisor(bl)
Effective cartier divisor
on scheme over QQ covered with 3 patches
1: [(s1//s0), (s2//s0), x] spec of quotient of multivariate polynomial ring
2: [(s0//s1), (s2//s1), y] spec of quotient of multivariate polynomial ring
3: [(s0//s2), (s1//s2), z] spec of quotient of multivariate polynomial ring
1: [(s1//s0), (s2//s0), x] V(0, 0, 0)
2: [(s0//s1), (s2//s1), y] V(0, 0, 0)
3: [(s0//s2), (s1//s2), z] V(0, 0, 0)
defined by
sheaf of ideals with restrictions
1: ideal(x)
Expand All @@ -67,7 +67,7 @@ defined by
julia> Z = center(bl)
Sheaf of ideals
on scheme over QQ covered with 1 patch
1: [x, y, z] spec of multivariate polynomial ring
1: [x, y, z] affine 3-space
with restriction
1: ideal(x, y, z)
```
Expand Down
12 changes: 6 additions & 6 deletions experimental/Schemes/CartierDivisor.jl
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Expand Up @@ -156,9 +156,9 @@ julia> II = IdealSheaf(Y, I);
julia> E = effective_cartier_divisor(II)
Effective cartier divisor
on scheme over QQ covered with 3 patches
1: [(y//x), (z//x)] spec of multivariate polynomial ring
2: [(x//y), (z//y)] spec of multivariate polynomial ring
3: [(x//z), (y//z)] spec of multivariate polynomial ring
1: [(y//x), (z//x)] affine 2-space
2: [(x//y), (z//y)] affine 2-space
3: [(x//z), (y//z)] affine 2-space
defined by
sheaf of ideals with restrictions
1: ideal(-(y//x)^2*(z//x) + 1)
Expand Down Expand Up @@ -222,9 +222,9 @@ julia> II = IdealSheaf(Y, I);
julia> effective_cartier_divisor(II)
Effective cartier divisor
on scheme over QQ covered with 3 patches
1: [(y//x), (z//x)] spec of multivariate polynomial ring
2: [(x//y), (z//y)] spec of multivariate polynomial ring
3: [(x//z), (y//z)] spec of multivariate polynomial ring
1: [(y//x), (z//x)] affine 2-space
2: [(x//y), (z//y)] affine 2-space
3: [(x//z), (y//z)] affine 2-space
defined by
sheaf of ideals with restrictions
1: ideal(-(y//x)^2*(z//x) + 1)
Expand Down
16 changes: 8 additions & 8 deletions experimental/Schemes/CoherentSheaves.jl
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Expand Up @@ -717,10 +717,10 @@ with homogeneous coordinates [s0, s1, s2, s3]
julia> twisting_sheaf(P, 4)
Coherent sheaf of modules
on scheme over QQ covered with 4 patches
1: [(s1//s0), (s2//s0), (s3//s0)] spec of multivariate polynomial ring
2: [(s0//s1), (s2//s1), (s3//s1)] spec of multivariate polynomial ring
3: [(s0//s2), (s1//s2), (s3//s2)] spec of multivariate polynomial ring
4: [(s0//s3), (s1//s3), (s2//s3)] spec of multivariate polynomial ring
1: [(s1//s0), (s2//s0), (s3//s0)] affine 3-space
2: [(s0//s1), (s2//s1), (s3//s1)] affine 3-space
3: [(s0//s2), (s1//s2), (s3//s2)] affine 3-space
4: [(s0//s3), (s1//s3), (s2//s3)] affine 3-space
with restrictions
1: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
2: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
Expand Down Expand Up @@ -778,10 +778,10 @@ with homogeneous coordinates [s0, s1, s2, s3]
julia> tautological_bundle(P)
Coherent sheaf of modules
on scheme over QQ covered with 4 patches
1: [(s1//s0), (s2//s0), (s3//s0)] spec of multivariate polynomial ring
2: [(s0//s1), (s2//s1), (s3//s1)] spec of multivariate polynomial ring
3: [(s0//s2), (s1//s2), (s3//s2)] spec of multivariate polynomial ring
4: [(s0//s3), (s1//s3), (s2//s3)] spec of multivariate polynomial ring
1: [(s1//s0), (s2//s0), (s3//s0)] affine 3-space
2: [(s0//s1), (s2//s1), (s3//s1)] affine 3-space
3: [(s0//s2), (s1//s2), (s3//s2)] affine 3-space
4: [(s0//s3), (s1//s3), (s2//s3)] affine 3-space
with restrictions
1: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
2: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
Expand Down
18 changes: 9 additions & 9 deletions experimental/Schemes/FunctionFields.jl
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Expand Up @@ -49,9 +49,9 @@ Scheme
over rational field
with default covering
described by patches
1: spec of quotient of multivariate polynomial ring
2: spec of quotient of multivariate polynomial ring
3: spec of quotient of multivariate polynomial ring
1: V(-(y//x)^2*(z//x) + 1)
2: V((x//y)^3 - (z//y))
3: V((x//z)^3 - (y//z)^2)
in the coordinate(s)
1: [(y//x), (z//x)]
2: [(x//y), (z//y)]
Expand All @@ -60,18 +60,18 @@ with default covering
julia> K = function_field(Ycov)
Field of rational functions
on scheme over QQ covered with 3 patches
1: [(y//x), (z//x)] spec of quotient of multivariate polynomial ring
2: [(x//y), (z//y)] spec of quotient of multivariate polynomial ring
3: [(x//z), (y//z)] spec of quotient of multivariate polynomial ring
1: [(y//x), (z//x)] V(-(y//x)^2*(z//x) + 1)
2: [(x//y), (z//y)] V((x//y)^3 - (z//y))
3: [(x//z), (y//z)] V((x//z)^3 - (y//z)^2)
represented by
patch 1: fraction field of multivariate polynomial ring

julia> one(K)
Rational function
on scheme over QQ covered with 3 patches
1: [(y//x), (z//x)] spec of quotient of multivariate polynomial ring
2: [(x//y), (z//y)] spec of quotient of multivariate polynomial ring
3: [(x//z), (y//z)] spec of quotient of multivariate polynomial ring
1: [(y//x), (z//x)] V(-(y//x)^2*(z//x) + 1)
2: [(x//y), (z//y)] V((x//y)^3 - (z//y))
3: [(x//z), (y//z)] V((x//z)^3 - (y//z)^2)
represented by
patch 1: 1
```
Expand Down
6 changes: 3 additions & 3 deletions experimental/Schemes/IdealSheaves.jl
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Expand Up @@ -37,9 +37,9 @@ with homogeneous coordinates [x, y, z]
julia> IdealSheaf(Y, I)
Sheaf of ideals
on scheme over QQ covered with 3 patches
1: [(y//x), (z//x)] spec of multivariate polynomial ring
2: [(x//y), (z//y)] spec of multivariate polynomial ring
3: [(x//z), (y//z)] spec of multivariate polynomial ring
1: [(y//x), (z//x)] affine 2-space
2: [(x//y), (z//y)] affine 2-space
3: [(x//z), (y//z)] affine 2-space
with restrictions
1: ideal(-(y//x)^2*(z//x) + 1)
2: ideal((x//y)^3 - (z//y))
Expand Down
14 changes: 7 additions & 7 deletions experimental/Schemes/MorphismFromRationalFunctions.jl
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Expand Up @@ -12,8 +12,8 @@ Scheme
over rational field
with default covering
described by patches
1: spec of multivariate polynomial ring
2: spec of multivariate polynomial ring
1: affine 1-space
2: affine 1-space
in the coordinate(s)
1: [(t//s)]
2: [(s//t)]
Expand All @@ -23,9 +23,9 @@ Scheme
over rational field
with default covering
described by patches
1: spec of multivariate polynomial ring
2: spec of multivariate polynomial ring
3: spec of multivariate polynomial ring
1: affine 2-space
2: affine 2-space
3: affine 2-space
in the coordinate(s)
1: [(y//x), (z//x)]
2: [(x//y), (z//y)]
Expand All @@ -50,8 +50,8 @@ julia> realizations = Oscar.realize_on_patch(Phi, U);

julia> realizations[3]
Morphism
from [(t//s)] spec of localized ring
to [(x//z), (y//z)] spec of multivariate polynomial ring
from [(t//s)] AA^1 \ V()
to [(x//z), (y//z)] affine 2-space
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given by the pullback function
(x//z) -> (t//s)^2
(y//z) -> (t//s)
Expand Down
12 changes: 6 additions & 6 deletions experimental/Schemes/Sheaves.jl
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Expand Up @@ -430,9 +430,9 @@ Scheme
over rational field
with default covering
described by patches
1: spec of quotient of multivariate polynomial ring
2: spec of quotient of multivariate polynomial ring
3: spec of quotient of multivariate polynomial ring
1: V(-(y//x)^2*(z//x) + 1)
2: V((x//y)^3 - (z//y))
3: V((x//z)^3 - (y//z)^2)
in the coordinate(s)
1: [(y//x), (z//x)]
2: [(x//y), (z//y)]
Expand All @@ -441,9 +441,9 @@ with default covering
julia> OO(Ycov)
Structure sheaf of rings of regular functions
on scheme over QQ covered with 3 patches
1: [(y//x), (z//x)] spec of quotient of multivariate polynomial ring
2: [(x//y), (z//y)] spec of quotient of multivariate polynomial ring
3: [(x//z), (y//z)] spec of quotient of multivariate polynomial ring
1: [(y//x), (z//x)] V(-(y//x)^2*(z//x) + 1)
2: [(x//y), (z//y)] V((x//y)^3 - (z//y))
3: [(x//z), (y//z)] V((x//z)^3 - (y//z)^2)
```
"""
@attr StructureSheafOfRings function OO(X::AbsCoveredScheme)
Expand Down
15 changes: 9 additions & 6 deletions experimental/Schemes/duValSing.jl
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Expand Up @@ -12,13 +12,14 @@ julia> I = ideal(R,[w,x^2+y^3+z^4])
ideal(w, x^2 + y^3 + z^4)

julia> Rq,_ = quo(R,I)
(Quotient of multivariate polynomial ring by ideal with 2 generators, Map from
(Quotient of multivariate polynomial ring by ideal(w, x^2 + y^3 + z^4), Map from
Multivariate polynomial ring in 4 variables over QQ to Rq defined by a julia-function with inverse)

julia> X = Spec(Rq)
Spectrum
of quotient
of multivariate polynomial ring in 4 variables over QQ
of multivariate polynomial ring in 4 variables x, y, z, w
over rational field
by ideal(w, x^2 + y^3 + z^4)

julia> has_du_val_singularities(X)
Expand Down Expand Up @@ -74,7 +75,7 @@ julia> I = ideal(R,[w,x^2+y^3+z^4])
ideal(w, x^2 + y^3 + z^4)

julia> Rq,_ = quo(R,I)
(Quotient of multivariate polynomial ring by ideal with 2 generators, Map from
(Quotient of multivariate polynomial ring by ideal(w, x^2 + y^3 + z^4), Map from
Multivariate polynomial ring in 4 variables over QQ to Rq defined by a julia-function with inverse)

julia> J = ideal(R,[x,y,z,w])
Expand All @@ -83,7 +84,8 @@ ideal(x, y, z, w)
julia> X = Spec(Rq)
Spectrum
of quotient
of multivariate polynomial ring in 4 variables over QQ
of multivariate polynomial ring in 4 variables x, y, z, w
over rational field
by ideal(w, x^2 + y^3 + z^4)

julia> is_du_val_singularity(X,J)
Expand Down Expand Up @@ -148,7 +150,7 @@ julia> I = ideal(R,[w,x^2+y^3+z^4])
ideal(w, x^2 + y^3 + z^4)

julia> Rq,_ = quo(R,I)
(Quotient of multivariate polynomial ring by ideal with 2 generators, Map from
(Quotient of multivariate polynomial ring by ideal(w, x^2 + y^3 + z^4), Map from
Multivariate polynomial ring in 4 variables over QQ to Rq defined by a julia-function with inverse)

julia> J = ideal(R,[x,y,z,w])
Expand All @@ -157,7 +159,8 @@ ideal(x, y, z, w)
julia> X = Spec(Rq)
Spectrum
of quotient
of multivariate polynomial ring in 4 variables over QQ
of multivariate polynomial ring in 4 variables x, y, z, w
over rational field
by ideal(w, x^2 + y^3 + z^4)

julia> decide_du_val_singularity(X,J)
Expand Down
3 changes: 2 additions & 1 deletion src/AlgebraicGeometry/Schemes/AffineAlgebraicSet/Objects/Constructors.jl
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Expand Up @@ -107,7 +107,8 @@ which the scheme theoretic intersection remembers. Therefore they are different.
julia> Z = intersect(X, Y) # a non reduced scheme
Spectrum
of quotient
of multivariate polynomial ring in 2 variables over QQ
of multivariate polynomial ring in 2 variables x, y
over rational field
by ideal(x^2 - y, y)

julia> Zred == Z
Expand Down
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