chap9.txt

  
  9 Examples and Tests
  
  
  9.1 Basic Commands
  
    Example  
    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.}}
  
    Example  
    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