# Morphisms of affine schemes

## Constructors

### General constructors

SpecMorMethod
SpecMor(X::AbsSpec, Y::AbsSpec, 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> SpecMor(X, Y, gens(OO(X)))
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::AbsSpec{<: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)
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::AbsSpec, Y::AbsSpec; 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 over QQ
by ideal(x1)

julia> f = inclusion_morphism(Y, X)
Morphism
from [x1, x2, x3]  spec of quotient of multivariate polynomial ring
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(x1)

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

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 over QQ
by ideal(x1)

julia> m1 = inclusion_morphism(Y, X)
Morphism
from [x1, x2, x3]  spec of quotient of multivariate polynomial ring
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)
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)
Morphism
from [x1, x2, x3]  spec of quotient of multivariate polynomial ring
to   [x1, x2, x3]  spec of quotient of multivariate polynomial ring
given by the pullback function
x1 -> 0
x2 -> x2
x3 -> x3

julia> compose(m3, compose(m1, m2)) == m1
true
source
restrictMethod
restrict(f::SpecMor, U::AbsSpec, V::AbsSpec)

This method restricts the domain of the morphism $f$ to $U$ and its codomain to $V$.

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 over QQ
by ideal(x1)

julia> restrict(identity_map(X), Y, Y) == identity_map(Y)
true
source

## Attributes

### General attributes

domainMethod
domain(f::AbsSpecMor)

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 over QQ
by ideal(x1)

julia> f = inclusion_morphism(X, Y)
Morphism
from [x1, x2, x3]  spec of quotient of multivariate polynomial ring
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 over QQ
by ideal(x1)
source
codomainMethod
codomain(f::AbsSpecMor)

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 over QQ
by ideal(x1)

julia> f = inclusion_morphism(X, Y)
Morphism
from [x1, x2, x3]  spec of quotient of multivariate polynomial ring
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::AbsSpecMor)

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 over QQ
by ideal(x1)

julia> pullback(inclusion_morphism(X, Y))
Map with following data
Domain:
=======
Multivariate polynomial ring in 3 variables over QQ
Codomain:
=========
Quotient of multivariate polynomial ring by ideal with 1 generator
source
graphMethod
graph(f::AbsSpecMor)

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 over QQ
by ideal(x1)

julia> f = inclusion_morphism(X, Y)
Morphism
from [x1, x2, x3]  spec of quotient of multivariate polynomial ring
to   [x1, x2, x3]  affine 3-space over QQ
given by the pullback function
x1 -> 0
x2 -> x2
x3 -> x3

julia> graph(f)
(Spec of quotient of multivariate polynomial ring, Morphism: spec of quotient of multivariate polynomial ring -> spec of quotient of multivariate polynomial ring, Morphism: spec of quotient of multivariate polynomial ring -> affine 3-space over QQ with coordinates [x1, x2, x3])
source

### Special attributes

In addition to the standard getters and methods for instances of SpecMor, 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::AbsSpecMor, g::AbsSpecMor)

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

source
is_identity_mapMethod
is_identity_map(f::AbsSpecMor)

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 over QQ
by ideal(x1)

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

## Methods

fiber_productMethod
fiber_product(f::SpecMor{SpecType, SpecType, <:Any}, g::SpecMor{SpecType, SpecType, <:Any}) where {SpecType<:StdSpec}

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

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

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::AbsSpec{<: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 SimplifiedSpec $Y$ 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 ==.

source