Goto Chapter: Top 1 Ind

#### 1.1 GAP Categories

 ‣ IsGradedLeftOrRightPresentationMorphism( object ) ( filter )

Returns: true or false

The GAP category of morphisms in the category of graded left or right presentations.

 ‣ IsGradedLeftPresentationMorphism( object ) ( filter )

Returns: true or false

The GAP category of morphisms in the category of graded left presentations.

 ‣ IsGradedRightPresentationMorphism( object ) ( filter )

Returns: true or false

The GAP category of morphisms in the category of graded right presentations.

 ‣ IsGradedLeftOrRightPresentation( object ) ( filter )

Returns: true or false

The GAP category of objects in the category of graded left presentations or graded right presentations.

 ‣ IsGradedLeftPresentation( object ) ( filter )

Returns: true or false

The GAP category of objects in the category of graded left presentations.

 ‣ IsGradedRightPresentation( object ) ( filter )

Returns: true or false

The GAP category of objects in the category of graded right presentations.

#### 1.2 Constructors

 ‣ GradedPresentationMorphism( A, M, B ) ( operation )

Returns: a morphism in $$\mathrm{Hom}(A,B)$$

The arguments are an object $$A$$, a homalg matrix $$M$$, and another object $$B$$. $$A$$ and $$B$$ shall either both be objects in the category of graded left presentations or both be objects in the category of graded right presentations. The output is a morphism $$A \rightarrow B$$ in the the category of graded left or right presentations whose underlying matrix is given by $$M$$.

 ‣ AsGradedLeftPresentation( M ) ( attribute )

Returns: an object

The argument is a homalg matrix $$M$$ over a graded ring $$R$$. The output is an object in the category of graded left presentations over $$R$$. This object has $$M$$ as its underlying matrix.

 ‣ AsGradedRightPresentation( M ) ( attribute )

Returns: an object

The argument is a homalg matrix $$M$$ over a ring $$R$$. The output is an object in the category of right presentations over $$R$$. This object has $$M$$ as its underlying matrix.

 ‣ AsGradedLeftOrRightPresentation( M, l ) ( function )

Returns: an object

The arguments are a homalg matrix $$M$$ and a boolean $$l$$. If $$l$$ is true, the output is an object in the category of left presentations. If $$l$$ is false, the output is an object in the category of right presentations. In both cases, the underlying matrix of the result is $$M$$.

 ‣ GradedFreeLeftPresentation( r, R ) ( operation )

Returns: an object

The arguments are a non-negative integer $$r$$ and a graded homalg ring $$R$$. The output is an object in the category of graded left presentations over $$R$$. It is represented by the $$0 \times r$$ matrix and thus it is free of rank $$r$$.

 ‣ GradedFreeRightPresentation( r, R ) ( operation )

Returns: an object

The arguments are a non-negative integer $$r$$ and a graded homalg ring $$R$$. The output is an object in the category of graded right presentations over $$R$$. It is represented by the $$r \times 0$$ matrix and thus it is free of rank $$r$$.

##### 1.2-7 UnderlyingPresentationObject
 ‣ UnderlyingPresentationObject( A ) ( attribute )

Returns: a left or right presentation

The argument is an object $$A$$ in the category of graded left or right presentations over a homalg ring $$R$$. The output is the corresponding object in the category of left or right presentations.

##### 1.2-8 UnderlyingHomalgRing
 ‣ UnderlyingHomalgRing( A ) ( attribute )

Returns: a homalg ring

The argument is an object $$A$$ in the category of graded left or right presentations over a homalg ring $$R$$. The output is $$R$$.

##### 1.2-9 GeneratorDegrees
 ‣ GeneratorDegrees( A ) ( attribute )

Returns: a list

The argument is an object $$A$$ in the category of graded left of right presentations over a ring $$R$$. The output is a list of elements of the degree group of $$R$$, the weights of the generators of $$A$$.

##### 1.2-10 AffineDimension
 ‣ AffineDimension( A ) ( attribute )

Returns: an integer

Returns the Krull dimension (of the annihilator ideal) of the underlying nongraded module A. The underlying ring must be commutative.

 ‣ GradedLeftPresentations( R ) ( attribute )

Returns: a category

The argument is a graded homalg ring $$R$$. The output is the category of graded left presentations over $$R$$.

 ‣ GradedRightPresentations( R ) ( attribute )

Returns: a category

The argument is a graded homalg ring $$R$$. The output is the category of graded right presentations over $$R$$.

#### 1.3 Attributes

##### 1.3-1 UnderlyingHomalgRing
 ‣ UnderlyingHomalgRing( R ) ( attribute )

Returns: a homalg ring

The argument is a morphism $$\alpha$$ in the category of left or right presentations over a homalg ring $$R$$. The output is $$R$$.

##### 1.3-2 UnderlyingMatrix
 ‣ UnderlyingMatrix( alpha ) ( attribute )

Returns: a homalg matrix

The argument is a morphism $$\alpha$$ in the category of left or right presentations. The output is its underlying homalg matrix.

#### 1.4 Non-Categorical Operations

##### 1.4-1 StandardGeneratorMorphism
 ‣ StandardGeneratorMorphism( A, i ) ( operation )

Returns: a morphism in $$\mathrm{Hom}(F, A)$$

The argument is an object $$A$$ in the category of left or right presentations over a homalg ring $$R$$ with underlying matrix $$M$$ and an integer $$i$$. The output is a morphism $$F \rightarrow A$$ given by the $$i$$-th row or column of $$M$$, where $$F$$ is a free left or right presentation of rank $$1$$.

##### 1.4-2 CoverByProjective
 ‣ CoverByProjective( A ) ( attribute )

Returns: a morphism in $$\mathrm{Hom}(F,A)$$

The argument is an object $$A$$ in the category of left or right presentations. The output is a morphism from a free module $$F$$ to $$A$$, which maps the standard generators of the free module to the generators of $$A$$.

#### 1.5 Examples

gap> LoadPackage( "GradedModulePresentationsForCAP" );
true
gap> Q := HomalgFieldOfRationalsInSingular( );
Q
gap> S := GradedRing( Q["x,y"] );
Q[x,y]
(weights: yet unset)
gap> Sgrmod := GradedLeftPresentations( S );
The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])
gap> Display( Sgrmod );
A CAP category with name
The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]):

38 primitive operations were used to derive 228 operations for this category
which algorithmically
* IsMonoidalCategory
* IsAbelianCategoryWithEnoughProjectives
and not yet algorithmically
* IsSymmetricClosedMonoidalCategory
gap> #ListPrimitivelyInstalledOperationsOfCategory( Sgrmod );
gap> M := GradedFreeLeftPresentation( 2, S, [ 1, 1 ] );
<An object in The category of graded left f.p. modules over Q[x,y]
(with weights [ 1, 1 ])>
gap> N := GradedFreeLeftPresentation( 1, S, [ 0 ] );
<An object in The category of graded left f.p. modules over Q[x,y]
(with weights [ 1, 1 ])>
gap> mat := HomalgMatrix( "[x,y]", 2, 1, S );
<A 2 x 1 matrix over a graded ring>
gap> Display( mat );
x,
y
gap> phi := GradedPresentationMorphism( M, mat, N );
<A morphism in The category of graded left f.p. modules over Q[x,y]
(with weights [ 1, 1 ])>
gap> IsWellDefined( phi );
true
gap> IsMonomorphism( phi );
false
gap> IsEpimorphism( phi );
false
gap> iota := ImageEmbedding( phi );
<A monomorphism in The category of graded left f.p. modules over Q[x,y]
(with weights [ 1, 1 ])>
gap> IsMonomorphism( iota );
true
gap> IsIsomorphism( iota );
false
gap> coker_mod := CokernelObject( phi );
<An object in The category of graded left f.p. modules over Q[x,y]
(with weights [ 1, 1 ])>
gap> Display( coker_mod );
x,
y

An object in The category of graded left f.p. modules over Q[x,y]
(with weights [ 1, 1 ])

(graded, degree of generator:[ 0 ])
gap> IsZero( coker_mod );
false
gap> is_artinian := M -> AffineDimension( M ) <= 0;
function( M ) ... end
gap> C := FullSubcategoryByMembershipFunction( Sgrmod, is_artinian );
<Subcategory of The category of graded left f.p. modules over Q[x,y]
(with weights [ 1, 1 ]) by is_artinian>
gap> CohP1 := Sgrmod / C;
The Serre quotient category of The category of graded left f.p. modules
over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian
gap> Display( CohP1 );
A CAP category with name
The Serre quotient category of The category of graded left f.p. modules
over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian:

21 primitive operations were used to derive 187 operations for this category
which algorithmically
* IsAbelianCategory
gap> Sh := CanonicalProjection( CohP1 );
Localization functor of The Serre quotient category of The category of graded left
f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name:
is_artinian
gap> InstallFunctor( Sh, "Sheafification" );
gap> psi := ApplyFunctor( Sh, phi );
<A morphism in The Serre quotient category of The category of graded left
f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name:
is_artinian>
gap> IsMonomorphism( psi );
false
gap> IsEpimorphism( psi );
true
gap> coker_shv := CokernelObject( psi );
<A zero object in The Serre quotient category of The category of graded left
f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name:
is_artinian>
gap> IsZero( coker_shv );
true
gap> epsilon := ApplyFunctor( Sh, iota );
<A morphism in The Serre quotient category of The category of graded left
f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name:
is_artinian>
gap> IsIsomorphism( epsilon );
true

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