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### 9 Examples and Tests

#### 9.1 Basic Commands

gap> Q := HomalgFieldOfRationals();;
gap> A := VectorSpaceObject( 4, Q );;
gap> B := VectorSpaceObject( 3, Q );;
gap> C := VectorSpaceObject( 2, Q );;
gap> alpha := VectorSpaceMorphism( A,
> HomalgMatrix( [ [ 1, 1, 1 ], [ 0, 1, 1 ],
> [ 1, 0, 1 ], [ 1, 1, 0 ] ], 4, 3, Q ), B );;
gap> gamma := VectorSpaceMorphism( C,
> HomalgMatrix( [ [ -1, 1, -1 ], [ 1, 0, -1 ] ], 2, 3, Q ), B );;
gap> p := ProjectionInFactorOfFiberProduct( [ alpha, gamma ], 1 );;
gap> q := ProjectionInFactorOfFiberProduct( [ alpha, gamma ], 2 );;
gap> PreCompose( AsGeneralizedMorphism( alpha ), GeneralizedInverse( gamma ) );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> gen1 := PreCompose( AsGeneralizedMorphism( alpha ),
>                        GeneralizedInverse( gamma ) );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> gen2 := PreCompose( GeneralizedInverse( p ), AsGeneralizedMorphism( q ) );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> IsCongruentForMorphisms( gen1, gen2 );
true


#### 9.2 Intersection of Nodal Curve and Cusp

We are going to intersect the nodal curve $$f = y^2 - x^2(x+1)$$ and the cusp $$g = (x+y)^2 - (y-x)^3$$. The two curves are arranged in a way such that they intersect at $$(0,0)$$ with intersection number as high as possible. We are going to compute this intersection number using the definition of the intersection number as the length of the module $$R/(f,g)$$ localized at $$(0,0)$$. In order to model modules over the localization of $$Q[x,y]$$ at $$(0,0)$$, we use a suitable Serre quotient category. 1 2 1 1 true We are going to intersect the nodal curve $$f = y^2 - x^2(x+1)$$ and the cusp $$g = (x+y)^2 - (y-x)^3$$. The two curves are arranged in a way such that they intersect at $$(0,0)$$ with intersection number as high as possible. We are going to compute this intersection number using the definition of the intersection number as the length of the module $$R/(f,g)$$ localized at $$(0,0)$$. In order to model modules over the localization of $$Q[x,y]$$ at $$(0,0)$$, we use a suitable Serre quotient category. 1 2 1 1 true We are going to intersect the nodal curve $$f = y^2 - x^2(x+1)$$ and the cusp $$g = (x+y)^2 - (y-x)^3$$. The two curves are arranged in a way such that they intersect at $$(0,0)$$ with intersection number as high as possible. We are going to compute this intersection number using the definition of the intersection number as the length of the module $$R/(f,g)$$ localized at $$(0,0)$$. In order to model modules over the localization of $$Q[x,y]$$ at $$(0,0)$$, we use a suitable Serre quotient category. 1 2 1 1 true

#### 9.3 Sweep

$$\href{https://terrytao.wordpress.com/2015/10/07/sweeping-a-matrix-rotates-its-graph/}{\textrm{Geometric interpretation of sweeping a matrix by Terence Tao.}}$$

gap> Q := HomalgFieldOfRationals();;
gap> V := VectorSpaceObject( 3, Q );;
gap> mat := HomalgMatrix( [ [ 9, 8, 7 ], [ 6, 5, 4 ], [ 3, 2, 1 ] ], 3, 3, Q );;
gap> alpha := VectorSpaceMorphism( V, mat, V );;
gap> graph := FiberProductEmbeddingInDirectSum(
>             [ alpha, IdentityMorphism( V ) ] );;
gap> Display( graph );
[ [     1,    -2,     1,     0,     0,     0 ],
[  -4/3,   7/3,     0,     2,     1,     0 ],
[   5/3,  -8/3,     0,    -1,     0,     1 ] ]

A split mono morphism in Category of matrices over Q
gap> D := DirectSum( V, V );;
gap> rotmat := HomalgMatrix( [ [ 0, 0, 0, -1, 0, 0 ],
>                              [ 0, 1, 0, 0, 0, 0 ],
>                              [ 0, 0, 1, 0, 0, 0 ],
>                              [ 1, 0, 0, 0, 0, 0 ],
>                              [ 0, 0, 0, 0, 1, 0 ],
>                              [ 0, 0, 0, 0, 0, 1 ] ],
>                              6, 6, Q );;
gap> rot := VectorSpaceMorphism( D, rotmat, D );;
gap> p := PreCompose( graph, rot );;
gap> Display( p );
[ [     0,    -2,     1,    -1,     0,     0 ],
[     2,   7/3,     0,   4/3,     1,     0 ],
[    -1,  -8/3,     0,  -5/3,     0,     1 ] ]

A morphism in Category of matrices over Q
gap> pi1 := ProjectionInFactorOfDirectSum( [ V, V ], 1 );;
gap> pi2 := ProjectionInFactorOfDirectSum( [ V, V ], 2 );;
gap> reversed_arrow := PreCompose( p, pi1 );;
gap> arrow := PreCompose( p, pi2 );;
gap> g := GeneralizedMorphismBySpan( reversed_arrow, arrow );;
gap> IsHonest( g );
true
gap> sweep_1_alpha := HonestRepresentative( g );;
gap> Display( sweep_1_alpha );
[ [  -1/9,   8/9,   7/9 ],
[   2/3,  -1/3,  -2/3 ],
[   1/3,  -2/3,  -4/3 ] ]

A morphism in Category of matrices over Q
gap> Display( alpha );
[ [  9,  8,  7 ],
[  6,  5,  4 ],
[  3,  2,  1 ] ]

A morphism in Category of matrices over Q

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