GAP (GradedModulePresentationsForCAP) - Chapter 1: Graded Module Presentations

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

1.1-2 IsGradedLeftPresentationMorphism

1.1-3 IsGradedRightPresentationMorphism

1.1-4 IsGradedLeftOrRightPresentation

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

1.1-5 IsGradedLeftPresentation

1.1-6 IsGradedRightPresentation

1.2 Constructors

1.2-1 GradedPresentationMorphism

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.

1.2-2 AsGradedLeftPresentation

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.

1.2-3 AsGradedRightPresentation

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.

1.2-4 AsGradedLeftOrRightPresentation

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.

1.2-5 GradedFreeLeftPresentation

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.

1.2-6 GradedFreeRightPresentation

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

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

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

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

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

1.2-11 GradedLeftPresentations

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

1.2-12 GradedRightPresentations

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

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

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

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

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
(over a graded ring)
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
(over a graded ring)

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

1 Graded Module Presentations

1.1 GAP Categories

1.1-1 IsGradedLeftOrRightPresentationMorphism