GAP (LocalizeRingForHomalg) - Chapter 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 )

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

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

4.1-2 IsHomalgLocalRingElementRep

‣ IsHomalgLocalRingElementRep( r ) ( representation )

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

4.1-3 IsHomalgLocalMatrixRep

‣ IsHomalgLocalMatrixRep( A ) ( representation )

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

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

4.2 Rings: Attributes

4.2-1 GeneratorsOfMaximalLeftIdeal

‣ GeneratorsOfMaximalLeftIdeal( R ) ( attribute )

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 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 )

4.3-2 AssociatedGlobalRing

‣ AssociatedGlobalRing( r ) ( operation )

4.3-3 AssociatedGlobalRing

‣ AssociatedGlobalRing( mat ) ( operation )

4.3-4 Numerator

‣ Numerator( r ) ( operation )

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 )

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

4.3-6 Denominator

‣ Denominator( r ) ( operation )

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 )

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

4.3-8 Name

‣ Name( r ) ( operation )

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 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 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 )

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 )

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 )

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 )

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 )

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 )

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.

4 Localize Rings