GAP (LocalizeRingForHomalg) - Chapter 4: Localize Rings

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4 Localize Rings
4.1 Category and Representations
4.1-1 IsHomalgLocalRingRep
4.1-2 IsHomalgLocalRingElementRep
4.1-3 IsHomalgLocalMatrixRep
4.2 Rings: Attributes
4.2-1 GeneratorsOfMaximalLeftIdeal
4.2-2 GeneratorsOfMaximalRightIdeal
4.3 Operations and Functions
4.3-1 AssociatedGlobalRing
4.3-2 AssociatedGlobalRing
4.3-3 AssociatedGlobalRing
4.3-4 Numerator
4.3-5 Numerator
4.3-6 Denominator
4.3-7 Denominator
4.3-8 Name
4.3-9 SetMatElm
4.3-10 AddToMatElm
4.3-11 MatElmAsString
4.3-12 MatElm
4.3-13 Cancel
4.3-14 LocalizeAt
4.3-15 LocalizeAtZero
4.3-16 LocalizePolynomialRingAtZeroWithMora
4.3-17 HomalgLocalRingElement
4.3-18 HomalgLocalMatrix

### 4 Localize Rings

The package LocalizeRingForHomalg defines the classes of local(ized) rings, local ring elements and local matrices. These three objects can be used as data structures defined in MatricesForHomalg on which the homalg project can rely to do homological computations over localized rings.

A homalg local ring element contains two homalg ring elements, a numerator (--> `Numerator` (4.3-4)) and a denominator (--> `Denominator` (4.3-6)). A homalg local matrix contains a global homalg matrix as a numerator (--> `Numerator` (4.3-5)) and a ring element as a denominator (--> `Denominator` (4.3-7)). New constructors for ring elements and matrices are `HomalgLocalRingElement` (4.3-17) and `HomalgLocalMatrix` (4.3-18) in addition to the standard contructors introduced in other packages of the homalg project.

The local rings most prominently can be used with methods known from general homalg rings. The methods for doing the computations are presented in the appendix (A), since they are not for external use. The new attributes and operations are documented here.

Since the objects inplemented here are representations from objects elsewhere in the homalg project (i.e. MatricesForHomalg), we want to stress that there are many other operations in homalg, which can be used in connection with the ones presented here. A few of them can be found in the examples and the appendix of this documentation.

#### 4.1 Category and Representations

##### 4.1-1 IsHomalgLocalRingRep

`‣ IsHomalgLocalRingRep`( R ) | ( representation )

Returns: true or false

The representation of homalg local rings.

(It is a subrepresentation of the GAP representation
`IsHomalgRingOrFinitelyPresentedModuleRep`.)

    DeclareRepresentation( "IsHomalgLocalRingRep",
            IsHomalgRing
            and IsHomalgRingOrFinitelyPresentedModuleRep,
            [ "ring" ] );

##### 4.1-2 IsHomalgLocalRingElementRep

`‣ IsHomalgLocalRingElementRep`( r ) | ( representation )

Returns: true or false

The representation of elements of homalg local rings.

(It is a representation of the GAP category `IsHomalgRingElement`.)

    DeclareRepresentation( "IsHomalgLocalRingElementRep",
            IsHomalgRingElement,
            [ "pointer" ] );

##### 4.1-3 IsHomalgLocalMatrixRep

`‣ IsHomalgLocalMatrixRep`( A ) | ( representation )

Returns: true or false

The representation of homalg matrices with entries in a homalg local ring.

(It is a representation of the GAP category `IsHomalgMatrix`.)

    DeclareRepresentation( "IsHomalgLocalMatrixRep",
            IsHomalgMatrix,
            [ ] );

#### 4.2 Rings: Attributes

##### 4.2-1 GeneratorsOfMaximalLeftIdeal

`‣ GeneratorsOfMaximalLeftIdeal`( R ) | ( attribute )

Returns: a homalg matrix

Returns the generators of the maximal ideal, at which R was created. The generators are given as a column over the associated global ring.

##### 4.2-2 GeneratorsOfMaximalRightIdeal

`‣ GeneratorsOfMaximalRightIdeal`( R ) | ( attribute )

Returns: a homalg matrix

Returns the generators of the maximal ideal, at which R was created. The generators are given as a row over the associated global ring.

#### 4.3 Operations and Functions

##### 4.3-1 AssociatedGlobalRing

`‣ AssociatedGlobalRing`( R ) | ( operation )

Returns: a (global) homalg ring

The global homalg ring, from which the local ring R was created.

##### 4.3-2 AssociatedGlobalRing

`‣ AssociatedGlobalRing`( r ) | ( operation )

Returns: a (global) homalg ring

The global homalg ring, from which the local ring element r was created.

##### 4.3-3 AssociatedGlobalRing

`‣ AssociatedGlobalRing`( mat ) | ( operation )

Returns: a (global) homalg ring

The global homalg ring, from which the local matrix mat was created.

##### 4.3-4 Numerator

`‣ Numerator`( r ) | ( operation )

Returns: a (global) homalg ring element

The numerator from a local ring element r, which is a homalg ring element from the computation ring.

##### 4.3-5 Numerator

`‣ Numerator`( mat ) | ( operation )

Returns: a (global) homalg matrix

The numerator from a local matrix mat, which is a homalg matrix from the computation ring.

##### 4.3-6 Denominator

`‣ Denominator`( r ) | ( operation )

Returns: a (global) homalg ring element

The denominator from a local ring element r, which is a homalg ring element from the computation ring.

##### 4.3-7 Denominator

`‣ Denominator`( mat ) | ( operation )

Returns: a (global) homalg ring element

The denominator from a local matrix mat, which is a homalg matrix from the computation ring.

##### 4.3-8 Name

`‣ Name`( r ) | ( operation )

Returns: a string

The name of the local ring element r.

##### 4.3-9 SetMatElm

`‣ SetMatElm`( mat, i, j, r, R ) | ( operation )

Changes the entry (i,j) of the local matrix mat to the value r. Here R is the (local) homalg ring involved in these computations.

##### 4.3-10 AddToMatElm

`‣ AddToMatElm`( mat, i, j, r, R ) | ( operation )

Changes the entry (i,j) of the local matrix mat by adding the value r to it. Here R is the (local) homalg ring involved in these computations.

##### 4.3-11 MatElmAsString

`‣ MatElmAsString`( mat, i, j, R ) | ( operation )

Returns: a string

Returns the entry (i,j) of the local matrix mat as a string. Here R is the (local) homalg ring involved in these computations.

##### 4.3-12 MatElm

`‣ MatElm`( mat, i, j, R ) | ( operation )

Returns: a local ring element

Returns the entry (i,j) of the local matrix mat. Here R is the (local) homalg ring involved in these computations.

##### 4.3-13 Cancel

`‣ Cancel`( a, b ) | ( operation )

Returns: two ring elements

For \(\textit{a}=a'*c\) and \(\textit{b}=b'*c\) return \(a'\) and \(b'\). The exact form of \(c\) depends on whether a procedure for gcd computation is included in the ring package.

##### 4.3-14 LocalizeAt

`‣ LocalizeAt`( R, l ) | ( operation )

Returns: a local ring

If l is a list of elements of the global ring R generating a maximal ideal, the method creates the corresponding localization of R at the complement of the maximal ideal.

##### 4.3-15 LocalizeAtZero

`‣ LocalizeAtZero`( R ) | ( operation )

Returns: a local ring

This method creates the corresponding localization of R at the complement of the maximal ideal generated by the indeterminates ("at zero").

##### 4.3-16 LocalizePolynomialRingAtZeroWithMora

`‣ LocalizePolynomialRingAtZeroWithMora`( R ) | ( operation )

Returns: a local ring

This method localizes the ring R at zero and this localized ring is returned. The ring table uses Mora's algorithm as implemented Singular for low level computations.

##### 4.3-17 HomalgLocalRingElement

`‣ HomalgLocalRingElement`( numer, denom, R ) | ( function )

`‣ HomalgLocalRingElement`( numer, R ) | ( function )

Returns: a local ring element

Creates the local ring element \(\textit{numer}/\textit{denom}\) or in the second case \(\textit{numer}/1\) for the local ring R. Both numer and denom may either be a string describing a valid global ring element or from the global ring or computation ring.

##### 4.3-18 HomalgLocalMatrix

`‣ HomalgLocalMatrix`( numer, denom, R ) | ( function )

`‣ HomalgLocalMatrix`( numer, R ) | ( function )

Returns: a local matrix

Creates the local matrix \(\textit{numer}/\textit{denom}\) or in the second case \(\textit{numer}/1\) for the local ring R. Both numer and denom may either be from the global ring or the computation ring.

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