4 Localize Rings

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

4.3-1 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

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.

`‣ 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" ] );

`‣ 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" ] );

`‣ 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, [ ] );

`‣ 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.

`‣ 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.

`‣ AssociatedGlobalRing` ( R ) | ( operation ) |

Returns: a (global) **homalg** ring

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

`‣ AssociatedGlobalRing` ( r ) | ( operation ) |

Returns: a (global) **homalg** ring

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

`‣ AssociatedGlobalRing` ( mat ) | ( operation ) |

Returns: a (global) **homalg** ring

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

`‣ 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.

`‣ Numerator` ( mat ) | ( operation ) |

Returns: a (global) **homalg** matrix

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

`‣ 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.

`‣ 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.

`‣ Name` ( r ) | ( operation ) |

Returns: a string

The name of the local ring element `r`.

`‣ 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.

`‣ 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.

`‣ 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.

`‣ 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.

`‣ Cancel` ( a, b ) | ( operation ) |

Returns: two ring elements

For `a`=a'*c and `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.

`‣ 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.

`‣ 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").

`‣ 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.

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

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

Returns: a local ring element

Creates the local ring element `numer`/`denom` or in the second case `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.

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

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

Returns: a local matrix

Creates the local matrix `numer`/`denom` or in the second case `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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