GAP (MatricesForHomalg) - Chapter 4: Ring Maps

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4 Ring Maps
4.1 Ring Maps: Category and Representations
4.1-1 IsHomalgRingMap
4.1-2 IsHomalgRingSelfMap
4.1-3 IsHomalgRingMapRep
4.2 Ring Maps: Constructors
4.2-1 RingMap
4.3 Ring Maps: Properties
4.3-1 IsMorphism
4.3-2 IsIdentityMorphism
4.3-3 IsMonomorphism
4.3-4 IsEpimorphism
4.3-5 IsIsomorphism
4.3-6 IsAutomorphism
4.4 Ring Maps: Attributes
4.4-1 Source
4.4-2 Range
4.4-3 DegreeOfMorphism
4.4-4 CoordinateRingOfGraph
4.5 Ring Maps: Operations and Functions

### 4 Ring Maps

A homalg ring map is a data structure for maps between finitely generated rings. homalg more or less provides the basic declarations and installs the generic methods for ring maps, but it is up to other high level packages to install methods applicable to specific rings. For example, the package Sheaves provides methods for ring maps of (finitely generated) affine rings.

#### 4.1 Ring Maps: Category and Representations

##### 4.1-1 IsHomalgRingMap

`‣ IsHomalgRingMap`( phi ) | ( category )

Returns: `true` or `false`

The GAP category of ring maps.

##### 4.1-2 IsHomalgRingSelfMap

`‣ IsHomalgRingSelfMap`( phi ) | ( category )

Returns: `true` or `false`

The GAP category of ring self-maps.

(It is a subcategory of the GAP category `IsHomalgRingMap`.)

##### 4.1-3 IsHomalgRingMapRep

`‣ IsHomalgRingMapRep`( phi ) | ( representation )

Returns: `true` or `false`

The GAP representation of homalg ring maps.

(It is a representation of the GAP category `IsHomalgRingMap` (4.1-1).)

#### 4.2 Ring Maps: Constructors

##### 4.2-1 RingMap

`‣ RingMap`( images, S, T ) | ( operation )

Returns: a homalg ring map

This constructor returns a ring map (homomorphism) of finitely generated rings/algebras. It is represented by the images images of the set of generators of the source homalg ring S in terms of the generators of the target ring T (--> 3.2). Unless the source ring is free and given on free ring/algebra generators the returned map will cautiously be indicated using parenthesis: "homomorphism". To verify if the result is indeed a well defined map use `IsMorphism` (4.3-1). If source and target are identical objects, and only then, the ring map is created as a selfmap.

#### 4.3 Ring Maps: Properties

##### 4.3-1 IsMorphism

`‣ IsMorphism`( phi ) | ( property )

Returns: `true` or `false`

Check if phi is a well-defined map, i.e. independent of all involved presentations.

##### 4.3-2 IsIdentityMorphism

`‣ IsIdentityMorphism`( phi ) | ( property )

Returns: `true` or `false`

Check if the homalg ring map phi is the identity morphism.

##### 4.3-3 IsMonomorphism

`‣ IsMonomorphism`( phi ) | ( property )

Returns: `true` or `false`

Check if the homalg ring map phi is a monomorphism.

##### 4.3-4 IsEpimorphism

`‣ IsEpimorphism`( phi ) | ( property )

Returns: `true` or `false`

Check if the homalg ring map phi is an epimorphism.

##### 4.3-5 IsIsomorphism

`‣ IsIsomorphism`( phi ) | ( property )

Returns: `true` or `false`

Check if the homalg ring map phi is an isomorphism.

##### 4.3-6 IsAutomorphism

`‣ IsAutomorphism`( phi ) | ( property )

Returns: `true` or `false`

Check if the homalg ring map phi is an automorphism.

#### 4.4 Ring Maps: Attributes

##### 4.4-1 Source

`‣ Source`( phi ) | ( attribute )

Returns: a homalg ring

The source of the homalg ring map phi.

##### 4.4-2 Range

`‣ Range`( phi ) | ( attribute )

Returns: a homalg ring

The target (range) of the homalg ring map phi.

##### 4.4-3 DegreeOfMorphism

`‣ DegreeOfMorphism`( phi ) | ( attribute )

Returns: an integer

The degree of the morphism phi of graded rings.
(no method installed)

##### 4.4-4 CoordinateRingOfGraph

`‣ CoordinateRingOfGraph`( phi ) | ( attribute )

Returns: a homalg ring

The coordinate ring of the graph of the ring map phi.

#### 4.5 Ring Maps: Operations and Functions

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