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### 5 Homogeneous Matrices

The package GradedRingForHomalg defines the classes of graded rings, ring elements and homogeneous matrices over such rings. These three objects can be used as data structures defined in MatricesForHomalg on which the homalg project can rely to do homological computations over graded rings.

The graded rings most prominently can be used with methods known from general homalg rings. The methods for doing the computations are presented in the appendix (B), 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 MatricesForHomalg, 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.

#### 5.1 Homogeneous Matrices: Category and Representations

 ‣ IsHomalgMatrixOverGradedRingRep( A ) ( representation )

Returns: true or false

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

(It is a representation of the GAP category IsMatrixOverGradedRing.)

DeclareRepresentation( "IsHomalgMatrixOverGradedRingRep",
[ ] );


#### 5.2 Homogeneous Matrices: Constructors

 ‣ MatrixOverGradedRing( mat, S ) ( function )

Returns: a matrix over a graded ring

Creates a matrix for the graded ring S, where mat is a matrix over UnderlyingNonGradedRing(S).

#### 5.3 Homogeneous Matrices: Attributes

##### 5.3-1 DegreesOfEntries
 ‣ DegreesOfEntries( A ) ( attribute )

Returns: a listlist of degrees/multi-degrees

The matrix of degrees of the matrix A.

##### 5.3-2 NonTrivialDegreePerRow
 ‣ NonTrivialDegreePerRow( A[, col_degrees] ) ( attribute )

Returns: a list of degrees/multi-degrees

The list of first nontrivial degree per row of the matrix A.

##### 5.3-3 NonTrivialDegreePerColumn
 ‣ NonTrivialDegreePerColumn( A[, row_degrees] ) ( attribute )

Returns: a list of degrees/multi-degrees

The list of first nontrivial degree per column of the matrix A.

#### 5.4 Homogeneous Matrices: Operations and Functions

 ‣ UnderlyingNonGradedRing( mat ) ( operation )

Returns: a homalg ring

The nongraded ring underlying HomalgRing(mat).

##### 5.4-2 SetMatElm
 ‣ SetMatElm( mat, i, j, r, R ) ( operation )

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

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

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

##### 5.4-4 MatElmAsString
 ‣ MatElmAsString( mat, i, j, R ) ( operation )

Returns: a string

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

##### 5.4-5 MatElm
 ‣ MatElm( mat, i, j, R ) ( operation )