This corresponds to Example B.2 in [Bar09].
gap> zz := HomalgRingOfIntegers( ); Z gap> imat := HomalgMatrix( "[ \ > 262, -33, 75, -40, \ > 682, -86, 196, -104, \ > 1186, -151, 341, -180, \ > -1932, 248, -556, 292, \ > 1018, -127, 293, -156 \ > ]", 5, 4, zz ); <A 5 x 4 matrix over an internal ring> gap> M := LeftPresentation( imat ); <A left module presented by 5 relations for 4 generators> gap> N := Hom( zz, M ); <A rank 1 right module on 4 generators satisfying yet unknown relations> gap> F := InsertObjectInMultiFunctor( Functor_Hom_for_fp_modules, 2, N, "TensorN" ); <The functor TensorN for f.p. modules and their maps over computable rings> gap> G := LeftDualizingFunctor( zz );; gap> II_E := GrothendieckSpectralSequence( F, G, M ); <A stable homological spectral sequence with sheets at levels [ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x [ 0 .. 1 ]> gap> Display( II_E ); The associated transposed spectral sequence: a homological spectral sequence at bidegrees [ [ 0 .. 1 ], [ -1 .. 0 ] ] --------- Level 0: * * * * --------- Level 1: * * . . --------- Level 2: s s . . Now the spectral sequence of the bicomplex: a homological spectral sequence at bidegrees [ [ -1 .. 0 ], [ 0 .. 1 ] ] --------- Level 0: * * * * --------- Level 1: * * . s --------- Level 2: s s . s gap> filt := FiltrationBySpectralSequence( II_E, 0 ); <An ascending filtration with degrees [ -1 .. 0 ] and graded parts: 0: <A non-torsion left module presented by 3 relations for 4 generators> -1: <A non-zero left module presented by 21 relations for 8 generators> of <A non-zero left module presented by 31 relations for 19 generators>> gap> ByASmallerPresentation( filt ); <An ascending filtration with degrees [ -1 .. 0 ] and graded parts: 0: <A rank 1 left module presented by 2 relations for 3 generators> -1: <A non-zero torsion left module presented by 6 relations for 6 generators> of <A rank 1 left module presented by 8 relations for 9 generators>> gap> m := IsomorphismOfFiltration( filt ); <A non-zero isomorphism of left modules>
This corresponds to Example B.3 in [Bar09].
gap> zz := HomalgRingOfIntegers( ); Z gap> imat := HomalgMatrix( "[ \ > 262, -33, 75, -40, \ > 682, -86, 196, -104, \ > 1186, -151, 341, -180, \ > -1932, 248, -556, 292, \ > 1018, -127, 293, -156 \ > ]", 5, 4, zz ); <A 5 x 4 matrix over an internal ring> gap> M := LeftPresentation( imat ); <A left module presented by 5 relations for 4 generators> gap> filt := PurityFiltration( M ); <The ascending purity filtration with degrees [ -1 .. 0 ] and graded parts: 0: <A free left module of rank 1 on a free generator> -1: <A non-zero torsion left module presented by 2 relations for 2 generators> of <A non-pure rank 1 left module presented by 2 relations for 3 generators>> gap> M; <A non-pure rank 1 left module presented by 2 relations for 3 generators> gap> II_E := SpectralSequence( filt ); <A stable homological spectral sequence with sheets at levels [ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x [ 0 .. 1 ]> gap> Display( II_E ); The associated transposed spectral sequence: a homological spectral sequence at bidegrees [ [ 0 .. 1 ], [ -1 .. 0 ] ] --------- Level 0: * * * * --------- Level 1: * * . . --------- Level 2: s . . . Now the spectral sequence of the bicomplex: a homological spectral sequence at bidegrees [ [ -1 .. 0 ], [ 0 .. 1 ] ] --------- Level 0: * * * * --------- Level 1: * * . s --------- Level 2: s . . s gap> m := IsomorphismOfFiltration( filt ); <A non-zero isomorphism of left modules> gap> IsIdenticalObj( Range( m ), M ); true gap> Source( m ); <A non-torsion left module presented by 2 relations for 3 generators (locked)> gap> Display( last ); [ [ 0, 2, 0 ], [ 0, 0, 12 ] ] Cokernel of the map Z^(1x2) --> Z^(1x3), currently represented by the above matrix gap> Display( filt ); Degree 0: Z^(1 x 1) ---------- Degree -1: Z/< 2 > + Z/< 12 >
This corresponds to Example B.5 in [Bar09].
gap> zz := HomalgRingOfIntegers( ); Z gap> imat := HomalgMatrix( "[ \ > 262, -33, 75, -40, \ > 682, -86, 196, -104, \ > 1186, -151, 341, -180, \ > -1932, 248, -556, 292, \ > 1018, -127, 293, -156 \ > ]", 5, 4, zz ); <A 5 x 4 matrix over an internal ring> gap> M := LeftPresentation( imat ); <A left module presented by 5 relations for 4 generators> gap> F := InsertObjectInMultiFunctor( Functor_TensorProduct_for_fp_modules, 2, M, "TensorM" ); <The functor TensorM for f.p. modules and their maps over computable rings> gap> G := LeftDualizingFunctor( zz );; gap> II_E := GrothendieckSpectralSequence( F, G, M ); <A stable cohomological spectral sequence with sheets at levels [ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x [ 0 .. 1 ]> gap> Display( II_E ); The associated transposed spectral sequence: a cohomological spectral sequence at bidegrees [ [ 0 .. 1 ], [ -1 .. 0 ] ] --------- Level 0: * * * * --------- Level 1: * * . . --------- Level 2: s s . . Now the spectral sequence of the bicomplex: a cohomological spectral sequence at bidegrees [ [ -1 .. 0 ], [ 0 .. 1 ] ] --------- Level 0: * * * * --------- Level 1: * * . s --------- Level 2: s s . s gap> filt := FiltrationBySpectralSequence( II_E, 0 ); <A descending filtration with degrees [ -1 .. 0 ] and graded parts: -1: <A non-zero left module presented by yet unknown relations for 6 generator\ s> 0: <A non-zero left module presented by yet unknown relations for 4 generators\ > of <A left module presented by yet unknown relations for 14 generators>> gap> ByASmallerPresentation( filt ); <A descending filtration with degrees [ -1 .. 0 ] and graded parts: -1: <A non-zero torsion left module presented by 4 relations for 4 generators> 0: <A rank 1 left module presented by 2 relations for 3 generators> of <A rank 1 left module presented by 6 relations for 7 generators>> gap> m := IsomorphismOfFiltration( filt ); <A non-zero isomorphism of left modules>
This corresponds to Example B.6 in [Bar09].
gap> zz := HomalgRingOfIntegers( ); Z gap> imat := HomalgMatrix( "[ \ > 262, -33, 75, -40, \ > 682, -86, 196, -104, \ > 1186, -151, 341, -180, \ > -1932, 248, -556, 292, \ > 1018, -127, 293, -156 \ > ]", 5, 4, zz ); <A 5 x 4 matrix over an internal ring> gap> M := LeftPresentation( imat ); <A left module presented by 5 relations for 4 generators> gap> P := Resolution( M ); <A non-zero right acyclic complex containing a single morphism of left modules\ at degrees [ 0 .. 1 ]> gap> GP := Hom( P ); <A non-zero acyclic cocomplex containing a single morphism of right modules at\ degrees [ 0 .. 1 ]> gap> FGP := GP * P; <A non-zero acyclic cocomplex containing a single morphism of left complexes a\ t degrees [ 0 .. 1 ]> gap> BC := HomalgBicomplex( FGP ); <A non-zero bicocomplex containing left modules at bidegrees [ 0 .. 1 ]x [ -1 .. 0 ]> gap> p_degrees := ObjectDegreesOfBicomplex( BC )[1]; [ 0, 1 ] gap> II_E := SecondSpectralSequenceWithFiltration( BC, p_degrees ); <A stable cohomological spectral sequence with sheets at levels [ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x [ 0 .. 1 ]> gap> Display( II_E ); The associated transposed spectral sequence: a cohomological spectral sequence at bidegrees [ [ 0 .. 1 ], [ -1 .. 0 ] ] --------- Level 0: * * * * --------- Level 1: * * . . --------- Level 2: s s . . Now the spectral sequence of the bicomplex: a cohomological spectral sequence at bidegrees [ [ -1 .. 0 ], [ 0 .. 1 ] ] --------- Level 0: * * * * --------- Level 1: * * * * --------- Level 2: s s . s gap> filt := FiltrationBySpectralSequence( II_E, 0 ); <A descending filtration with degrees [ -1 .. 0 ] and graded parts: -1: <A non-zero torsion left module presented by yet unknown relations for 4 generators> 0: <A rank 1 left module presented by 3 relations for 4 generators> of <A left module presented by yet unknown relations for 13 generators>> gap> ByASmallerPresentation( filt ); <A descending filtration with degrees [ -1 .. 0 ] and graded parts: -1: <A non-zero torsion left module presented by 4 relations for 4 generators> 0: <A rank 1 left module presented by 2 relations for 3 generators> of <A rank 1 left module presented by 6 relations for 7 generators>> gap> m := IsomorphismOfFiltration( filt ); <A non-zero isomorphism of left modules>
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