GAP (GradedRingForHomalg) - Appendix B: Overview of the GradedRingForHomalg Package Source Code

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B Overview of the GradedRingForHomalg Package Source Code
B.1 The generic Methods
B.1-1 BasisOfRowModule
B.1-2 DecideZeroRows
B.1-3 SyzygiesGeneratorsOfRows
B.2 Tools

### B Overview of the GradedRingForHomalg Package Source Code

This appendix is included in the documentation to shine some light on the mathematical backgrounds of this Package. Neither is it needed to work with this package nor should the methods presented here be called directly. The functions documented here are entries of the so called ring table and not to be called directly. There are higher level methods in declared and installed in MatricesForHomalg, which call this functions (--> `?MatricesForHomalg:The Basic Matrix Operations`).

#### B.1 The generic Methods

We will present some methods as an example, to show the idea:

##### B.1-1 BasisOfRowModule

`‣ BasisOfRowModule`( M ) | ( function )

Returns: a distinguished basis (i.e. a distinguished generating set) of the module generated by M

    BasisOfRowModule :=
      function( M )
         return MatrixOverGradedRing(
                        BasisOfRowModule( UnderlyingMatrixOverNonGradedRing( M ) ),
                        HomalgRing( M ) );
      end,

##### B.1-2 DecideZeroRows

`‣ DecideZeroRows`( A, B ) | ( function )

Returns: a reduced form of A with respect to B

    DecideZeroRows :=
      function( A, B )
        return MatrixOverGradedRing(
                       DecideZeroRows( UnderlyingMatrixOverNonGradedRing( A ),
                               UnderlyingMatrixOverNonGradedRing( B ) ),
                       HomalgRing( A ) );
      end,

##### B.1-3 SyzygiesGeneratorsOfRows

`‣ SyzygiesGeneratorsOfRows`( M ) | ( function )

Returns: a distinguished basis of the syzygies of the argument

    SyzygiesGeneratorsOfRows :=
      function( M )
        return MatrixOverGradedRing(
                       SyzygiesGeneratorsOfRows( UnderlyingMatrixOverNonGradedRing( M ) ),
                       HomalgRing( M ) );
      end,

#### B.2 Tools

The package GradedRingForHomalg also implements tool functions. These are referred to from MatricesForHomalg automatically. We list the implemented methods here are and refer to the MatricesForHomalg documentation (--> `?MatricesForHomalg: The Matrix Tool Operations` and `?MatricesForHomalg:RingElement`) for details. All tools functions from MatricesForHomalg not listed here are also supported by fallback tools.

- IsZero

- IsOne

- Minus

- DivideByUnit

- IsUnit

- Sum

- Product

- ShallowCopy

- ZeroMatrix

- IdentityMatrix

- AreEqualMatrices

- Involution

- TransposedMatrix

- CertainRows

- CertainColumns

- UnionOfRows

- UnionOfColumns

- DiagMat

- KroneckerMat

- DualKroneckerMat

- MulMat

- AddMat

- SubMat

- Compose

- NrRows

- NrColumns

- IsZeroMatrix

- IsDiagonalMatrix

- ZeroRows

- ZeroColumns

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