Quelle TorExt_Grothendieck.g Sprache: unbekannt

##  <#GAPDoc Label="TorExt-Grothendieck">
##  <Subsection Label="TorExt-Grothendieck">
##  <Heading>TorExt-Grothendieck</Heading>
##  This is Example B.5 in <Cite Key="BaSF"/>.
##  <Example><![CDATA[
##  gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
##  Q[x,y,z]
##  gap> wmat := HomalgMatrix( "[ \
##  > x*y,  y*z,    z,        0,         0,    \
##  > x^3*z,x^2*z^2,0,        x*z^2,     -z^2, \
##  > x^4,  x^3*z,  0,        x^2*z,     -x*z, \
##  > 0,    0,      x*y,      -y^2,      x^2-1,\
##  > 0,    0,      x^2*z,    -x*y*z,    y*z,  \
##  > 0,    0,      x^2*y-x^2,-x*y^2+x*y,y^2-y \
##  > ]", 6, 5, Qxyz );
##  <A 6 x 5 matrix over an external ring>
##  gap> W := LeftPresentation( wmat );
##  <A left module presented by 6 relations for 5 generators>
##  gap> F := InsertObjectInMultiFunctor( Functor_TensorProduct_for_fp_modules, 2, W, "TensorW" );
##  <The functor TensorW for f.p. modules and their maps over computable rings>
##  gap> G := LeftDualizingFunctor( Qxyz );;
##  gap> II_E := GrothendieckSpectralSequence( F, G, W );
##  <A stable cohomological spectral sequence with sheets at levels
##  [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
##  [ 0 .. 3 ]>
##  gap> Display( II_E );
##  The associated transposed spectral sequence:
##
##  a cohomological spectral sequence at bidegrees
##  [ [ 0 .. 3 ], [ -3 .. 0 ] ]
##  ---------
##  Level 0:
##
##   * * * *
##   * * * *
##   . * * *
##   . . * *
##  ---------
##  Level 1:
##
##   * * * *
##   . . . .
##   . . . .
##   . . . .
##  ---------
##  Level 2:
##
##   s s s s
##   . . . .
##   . . . .
##   . . . .
##
##  Now the spectral sequence of the bicomplex:
##
##  a cohomological spectral sequence at bidegrees
##  [ [ -3 .. 0 ], [ 0 .. 3 ] ]
##  ---------
##  Level 0:
##
##   * * * *
##   * * * *
##   . * * *
##   . . * *
##  ---------
##  Level 1:
##
##   * * * *
##   * * * *
##   . * * *
##   . . . *
##  ---------
##  Level 2:
##
##   * * s s
##   * * * *
##   . * * *
##   . . . *
##  ---------
##  Level 3:
##
##   * s s s
##   . s s s
##   . . s *
##   . . . s
##  ---------
##  Level 4:
##
##   s s s s
##   . s s s
##   . . s s
##   . . . s
##  gap> filt := FiltrationBySpectralSequence( II_E, 0 );
##  <A descending filtration with degrees [ -3 .. 0 ] and graded parts:
##
##  -3:   <A non-zero cyclic torsion left module presented by yet unknown relations \
##  for a cyclic generator>
##    -2:   <A non-zero left module presented by 17 relations for 6 generators>
##    -1:   <A non-zero left module presented by 27 relations for 12 generators>
##     0:   <A non-zero left module presented by 13 relations for 10 generators>
##  of
##  <A left module presented by yet unknown relations for 49 generators>>
##  gap> ByASmallerPresentation( filt );
##  <A descending filtration with degrees [ -3 .. 0 ] and graded parts:
##
##  -3:   <A non-zero cyclic torsion left module presented by 3 relations for a cycl\
##  ic generator>
##    -2:   <A non-zero left module presented by 12 relations for 4 generators>
##    -1:   <A non-zero left module presented by 21 relations for 8 generators>
##     0:   <A non-zero left module presented by 11 relations for 10 generators>
##  of
##  <A non-zero left module presented by 27 relations for 14 generators>>
##  gap> m := IsomorphismOfFiltration( filt );
##  <A non-zero isomorphism of left modules>
##  ]]></Example>
##  </Subsection>
##  <#/GAPDoc>

ReadPackage( "ExamplesForHomalg", "examples/ReducedBasisOfModule.g" );

InsertObjectInMultiFunctor( Functor_TensorProduct_for_fp_modules, 2, W, "TensorW" );

II_E := GrothendieckSpectralSequence( Functor_TensorW_for_fp_modules, LeftDualizingFunctor( Qxyz ), W );

filt := FiltrationBySpectralSequence( II_E );

ByASmallerPresentation( filt );

m := IsomorphismOfFiltration( filt );

#Display( StringTime( homalgTime( Qxyz ) ) );

Quelle TorExt_Grothendieck.g Sprache: unbekannt

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