GAP (GeneralizedMorphismsForCAP) - Chapter 9: Examples and Tests

Goto Chapter: Top 1 2 3 4 5 6 7 8 9 Ind
[Top of Book] [Contents] [Previous Chapter] [Next Chapter]

[MathJax off]

9 Examples and Tests
9.1 Basic Commands
9.2 Intersection of Nodal Curve and Cusp
9.3 Sweep

### 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 monomorphism 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

[Top of Book] [Contents] [Previous Chapter] [Next Chapter]
Goto Chapter: Top 1 2 3 4 5 6 7 8 9 Ind

generated by GAPDoc2HTML