# Morphisms of affine schemes

## Constructors

### General constructors

morphismMethod
morphism(X::AbsAffineScheme, Y::AbsAffineScheme, f::Vector{<:RingElem}; check::Bool=true)

This method constructs a morphism from the scheme $X$ to the scheme $Y$. For this one has to specify the images of the coordinates (the generators of ambient_coordinate_ring(Y)) under the pullback map $𝒪(Y) → 𝒪(X)$ as third argument.

Note that expensive checks can be turned off by setting check=false.

Examples

julia> X = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> Y = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> morphism(X, Y, gens(OO(X)))
Affine scheme morphism
from [x1, x2, x3]  affine 3-space over QQ
to   [x1, x2, x3]  affine 3-space over QQ
given by the pullback function
x1 -> x1
x2 -> x2
x3 -> x3
source

### Special constructors

identity_mapMethod
identity_map(X::AbsAffineScheme{<:Any, <:MPolyRing})

This method constructs the identity morphism from an affine scheme to itself.

Examples

julia> X = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> identity_map(X)
Affine scheme morphism
from [x1, x2, x3]  affine 3-space over QQ
to   [x1, x2, x3]  affine 3-space over QQ
given by the pullback function
x1 -> x1
x2 -> x2
x3 -> x3
source
inclusion_morphismMethod
inclusion_morphism(X::AbsAffineScheme, Y::AbsAffineScheme; check::Bool=true)

Return the inclusion map from $X$ to $Y$.

Examples

julia> X = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> R = OO(X)
Multivariate polynomial ring in 3 variables x1, x2, x3
over rational field

julia> (x1,x2,x3) = gens(R)
3-element Vector{QQMPolyRingElem}:
x1
x2
x3

julia> Y = subscheme(X, x1)
Spectrum
of quotient
of multivariate polynomial ring in 3 variables x1, x2, x3
over rational field
by ideal (x1)

julia> f = inclusion_morphism(Y, X)
Affine scheme morphism
from [x1, x2, x3]  scheme(x1)
to   [x1, x2, x3]  affine 3-space over QQ
given by the pullback function
x1 -> 0
x2 -> x2
x3 -> x3

julia> I = kernel(pullback(f))  # this is a way to obtain the ideal I ⊆  O(X) cutting out Y from X.
Ideal generated by
x1

julia> base_ring(I) == OO(X)
true
source
composeMethod
compose(f::AbsAffineSchemeMor, g::AbsAffineSchemeMor)

This method computes the composition of two morphisms.

Examples

julia> X = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> R = OO(X)
Multivariate polynomial ring in 3 variables x1, x2, x3
over rational field

julia> (x1,x2,x3) = gens(R)
3-element Vector{QQMPolyRingElem}:
x1
x2
x3

julia> Y = subscheme(X, x1)
Spectrum
of quotient
of multivariate polynomial ring in 3 variables x1, x2, x3
over rational field
by ideal (x1)

julia> m1 = inclusion_morphism(Y, X)
Affine scheme morphism
from [x1, x2, x3]  scheme(x1)
to   [x1, x2, x3]  affine 3-space over QQ
given by the pullback function
x1 -> 0
x2 -> x2
x3 -> x3

julia> m2 = identity_map(X)
Affine scheme morphism
from [x1, x2, x3]  affine 3-space over QQ
to   [x1, x2, x3]  affine 3-space over QQ
given by the pullback function
x1 -> x1
x2 -> x2
x3 -> x3

julia> m3 = identity_map(Y)
Affine scheme morphism
from [x1, x2, x3]  scheme(x1)
to   [x1, x2, x3]  scheme(x1)
given by the pullback function
x1 -> 0
x2 -> x2
x3 -> x3

julia> compose(m3, compose(m1, m2)) == m1
true
source
restrictMethod
restrict(f::AbsAffineSchemeMor, D::AbsAffineScheme, Z::AbsAffineScheme; check::Bool=true)

This method restricts the domain of the morphism $f$ to $D$ and its codomain to $Z$.

Examples

julia> X = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> R = OO(X)
Multivariate polynomial ring in 3 variables x1, x2, x3
over rational field

julia> (x1,x2,x3) = gens(R)
3-element Vector{QQMPolyRingElem}:
x1
x2
x3

julia> Y = subscheme(X, x1)
Spectrum
of quotient
of multivariate polynomial ring in 3 variables x1, x2, x3
over rational field
by ideal (x1)

julia> restrict(identity_map(X), Y, Y) == identity_map(Y)
true
source
restrict(f::SchemeMor, U::Scheme, V::Scheme; check::Bool=true)

Return the restriction $g: U → V$ of $f$ to $U$ and $V$.

source

## Attributes

### General attributes

domainMethod
domain(f::AbsAffineSchemeMor)

On a morphism $f : X → Y$ of affine schemes, this returns $X$.

Examples

julia> Y = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> R = OO(Y)
Multivariate polynomial ring in 3 variables x1, x2, x3
over rational field

julia> (x1,x2,x3) = gens(R)
3-element Vector{QQMPolyRingElem}:
x1
x2
x3

julia> X = subscheme(Y, x1)
Spectrum
of quotient
of multivariate polynomial ring in 3 variables x1, x2, x3
over rational field
by ideal (x1)

julia> f = inclusion_morphism(X, Y)
Affine scheme morphism
from [x1, x2, x3]  scheme(x1)
to   [x1, x2, x3]  affine 3-space over QQ
given by the pullback function
x1 -> 0
x2 -> x2
x3 -> x3

julia> domain(f)
Spectrum
of quotient
of multivariate polynomial ring in 3 variables x1, x2, x3
over rational field
by ideal (x1)
source
codomainMethod
codomain(f::AbsAffineSchemeMor)

On a morphism $f : X → Y$ of affine schemes, this returns $Y$.

Examples

julia> Y = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> R = OO(Y)
Multivariate polynomial ring in 3 variables x1, x2, x3
over rational field

julia> (x1,x2,x3) = gens(R)
3-element Vector{QQMPolyRingElem}:
x1
x2
x3

julia> X = subscheme(Y, x1)
Spectrum
of quotient
of multivariate polynomial ring in 3 variables x1, x2, x3
over rational field
by ideal (x1)

julia> f = inclusion_morphism(X, Y)
Affine scheme morphism
from [x1, x2, x3]  scheme(x1)
to   [x1, x2, x3]  affine 3-space over QQ
given by the pullback function
x1 -> 0
x2 -> x2
x3 -> x3

julia> codomain(f)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]
source
pullbackMethod
pullback(f::AbsAffineSchemeMor)

On a morphism $f : X → Y$ of affine schemes $X = Spec(S)$ and $Y = Spec(R)$, this returns the ring homomorphism $f^* : R → S$.

Examples

julia> Y = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> R = OO(Y)
Multivariate polynomial ring in 3 variables x1, x2, x3
over rational field

julia> (x1,x2,x3) = gens(R)
3-element Vector{QQMPolyRingElem}:
x1
x2
x3

julia> X = subscheme(Y, x1)
Spectrum
of quotient
of multivariate polynomial ring in 3 variables x1, x2, x3
over rational field
by ideal (x1)

julia> pullback(inclusion_morphism(X, Y))
Ring homomorphism
from multivariate polynomial ring in 3 variables over QQ
to quotient of multivariate polynomial ring by ideal (x1)
defined by
x1 -> 0
x2 -> x2
x3 -> x3
source
graphMethod
graph(f::AbsAffineSchemeMor)

Return the graph of $f : X → Y$ as a subscheme of $X×Y$ as well as the two projections to $X$ and $Y$.

Examples

julia> Y = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> R = OO(Y)
Multivariate polynomial ring in 3 variables x1, x2, x3
over rational field

julia> (x1,x2,x3) = gens(R)
3-element Vector{QQMPolyRingElem}:
x1
x2
x3

julia> X = subscheme(Y, x1)
Spectrum
of quotient
of multivariate polynomial ring in 3 variables x1, x2, x3
over rational field
by ideal (x1)

julia> f = inclusion_morphism(X, Y)
Affine scheme morphism
from [x1, x2, x3]  scheme(x1)
to   [x1, x2, x3]  affine 3-space over QQ
given by the pullback function
x1 -> 0
x2 -> x2
x3 -> x3

julia> graph(f)
(scheme(x1, -x1, x2 - x2, x3 - x3), Hom: scheme(x1, -x1, x2 - x2, x3 - x3) -> scheme(x1), Hom: scheme(x1, -x1, x2 - x2, x3 - x3) -> affine 3-space over QQ with coordinates [x1, x2, x3])
source
graph(TropC::TropicalCurve{minOrMax,false})

Return the graph of an abstract tropical curve TropC. Same as polyhedral_complex(tc).

source

### Special attributes

In addition to the standard getters and methods for instances of AffineSchemeMor, we also have

image_idealMethod
image_ideal(f::ClosedEmbedding)

For a closed embedding $f : X → Y$ of affine schemes $X = Spec(S)$ into $Y = Spec(R)$ such that $S ≅ R/I$ via $f$ for some ideal $I ⊂ R$ this returns $I$.

source

### Undocumented

The following functions do exist but are currently undocumented:

• underlying_morphism,
• complement_ideal,
• complement_scheme,
• preimage,
• inverse,
• various type getters.

## Properties

is_inverse_ofMethod
is_inverse_of(f::AbsAffineSchemeMor, g::AbsAffineSchemeMor)

This method checks if a morphism $f$ is the inverse of a morphism $g$.

source
is_identity_mapMethod
is_identity_map(f::AbsAffineSchemeMor)

This method checks if a morphism is the identity map.

Examples

julia> X = affine_space(QQ,3)
Affine space of dimension 3
over rational field
with coordinates [x1, x2, x3]

julia> R = OO(X)
Multivariate polynomial ring in 3 variables x1, x2, x3
over rational field

julia> (x1,x2,x3) = gens(R)
3-element Vector{QQMPolyRingElem}:
x1
x2
x3

julia> Y = subscheme(X, x1)
Spectrum
of quotient
of multivariate polynomial ring in 3 variables x1, x2, x3
over rational field
by ideal (x1)

julia> is_identity_map(inclusion_morphism(Y, X))
false
source

## Methods

fiber_productMethod
fiber_product(f::AbsAffineSchemeMor, g::AbsAffineSchemeMor)

For morphisms $f : X → Z$ and $g : Y → Z$ return the fiber product $X×Y$ over $Z$ together with its two canonical projections.

Whenever you have another set of maps a: W → X and b : W → Y forming a commutative square with f and g, you can use induced_map_to_fiber_product to create the resulting map W → X×Y.

source
productMethod
product(X::AbsAffineScheme, Y::AbsAffineScheme)

Return a triple $(X×Y, p₁, p₂)$ consisting of the product $X×Y$ over the common base ring $𝕜$ and the two projections $p₁ : X×Y → X$ and $p₂ : X×Y → Y$.

source
simplifyMethod
simplify(X::AbsAffineScheme{<:Field})

Given an affine scheme $X$ with coordinate ring $R = 𝕜[x₁,…,xₙ]/I$ (or a localization thereof), use Singular's elimpart to try to eliminate variables $xᵢ$ to arrive at a simpler presentation $R ≅ R' = 𝕜[y₁,…,yₘ]/J$ for some ideal $J$; return a SimplifiedAffineScheme $Y$ with $X$ as its original.

***Note:*** The ambient_coordinate_ring of the output Y will be different from the one of X and hence the two schemes will not compare using ==.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source