CapAndHomalg.jl
Examples
The following examples tests the functionality of the software projects
julia> using CapAndHomalg
julia> LoadPackage( "GradedModules" )
julia> ℚ = HomalgFieldOfRationalsInSingular( )
GAP: Q
julia> R = ℚ["x,y,z"]
GAP: Q[x,y,z]
julia> m = "[ 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 ]";
julia> m = HomalgMatrix( m, 6, 5, R )
GAP: <A 6 x 5 matrix over an external ring>
julia> M = LeftPresentation( m )
GAP: <A left module presented by 6 relations for 5 generators>
julia> Display( M )
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
Cokernel of the map
Q[x,y,z]^(1x6) --> Q[x,y,z]^(1x5),
currently represented by the above matrix
julia> filt = PurityFiltration( M )
GAP: <The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts:
0: <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4 generators>
-1: <A codegree-1-pure grade 1 left module presented by 4 relations for 3 generators>
-2: <A cyclic reflexively pure grade 2 left module presented by 2 relations for a cyclic generator>
-3: <A cyclic reflexively pure grade 3 left module presented by 3 relations for a cyclic generator>
of
<A non-pure rank 2 left module presented by 6 relations for 5 generators>>
julia> Display( filt )
Degree 0:
0, 0,x, -y,
x*y,0,-z,0,
x^2,0,0, -z
Cokernel of the map
Q[x,y,z]^(1x3) --> Q[x,y,z]^(1x4),
currently represented by the above matrix
----------
Degree -1:
y,-z,0,
0,x, -y,
x,0, -z,
0,-y,x^2-1
Cokernel of the map
Q[x,y,z]^(1x4) --> Q[x,y,z]^(1x3),
currently represented by the above matrix
----------
Degree -2:
Q[x,y,z]/< z, y-1 >
----------
Degree -3:
Q[x,y,z]/< z, y, x >
julia> II_E = SpectralSequence( filt )
GAP: <A stable homological spectral sequence with sheets at levels [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x[ 0 .. 3 ]>
julia> Display( II_E )
The associated transposed spectral sequence:
a homological spectral sequence at bidegrees
[ [ 0 .. 3 ], [ -3 .. 0 ] ]
---------
Level 0:
* * * *
* * * *
. * * *
. . * *
---------
Level 1:
* * * *
. . . .
. . . .
. . . .
---------
Level 2:
s . . .
. . . .
. . . .
. . . .
Now the spectral sequence of the bicomplex:
a homological spectral sequence at bidegrees
[ [ -3 .. 0 ], [ 0 .. 3 ] ]
---------
Level 0:
* * * *
* * * *
. * * *
. . * *
---------
Level 1:
* * * *
* * * *
. * * *
. . . *
---------
Level 2:
s . . .
* s . .
. * * .
. . . *
---------
Level 3:
s . . .
* s . .
. . s .
. . . *
---------
Level 4:
s . . .
. s . .
. . s .
. . . s
julia> FilteredByPurity( M )
GAP: <A non-pure rank 2 left module presented by 12 relations for 9 generators>
julia> Display( M )
0, 0,x, -y,0,1, 0, 0, 0,
x*y,0,-z,0, 0,0, 0, 0, 0,
x^2,0,0, -z,1,0, 0, 0, 0,
0, 0,0, 0, y,-z,0, 0, 0,
0, 0,0, 0, 0,x, -y, -1, 0,
0, 0,0, 0, x,0, -z, 0, -1,
0, 0,0, 0, 0,-y,x^2-1,0, 0,
0, 0,0, 0, 0,0, 0, z, 0,
0, 0,0, 0, 0,0, 0, y-1,0,
0, 0,0, 0, 0,0, 0, 0, z,
0, 0,0, 0, 0,0, 0, 0, y,
0, 0,0, 0, 0,0, 0, 0, x
Cokernel of the map
Q[x,y,z]^(1x12) --> Q[x,y,z]^(1x9),
currently represented by the above matrix
julia> OnFirstStoredPresentation( M )
GAP: <A non-pure rank 2 left module presented by 6 relations for 5 generators>
julia> Display( M )
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
Cokernel of the map
Q[x,y,z]^(1x6) --> Q[x,y,z]^(1x5),
currently represented by the above matrix
julia> OnLastStoredPresentation( M )
GAP: <A non-pure rank 2 left module presented by 12 relations for 9 generators>
julia> Display( M )
0, 0,x, -y,0,1, 0, 0, 0,
x*y,0,-z,0, 0,0, 0, 0, 0,
x^2,0,0, -z,1,0, 0, 0, 0,
0, 0,0, 0, y,-z,0, 0, 0,
0, 0,0, 0, 0,x, -y, -1, 0,
0, 0,0, 0, x,0, -z, 0, -1,
0, 0,0, 0, 0,-y,x^2-1,0, 0,
0, 0,0, 0, 0,0, 0, z, 0,
0, 0,0, 0, 0,0, 0, y-1,0,
0, 0,0, 0, 0,0, 0, 0, z,
0, 0,0, 0, 0,0, 0, 0, y,
0, 0,0, 0, 0,0, 0, 0, x
Cokernel of the map
Q[x,y,z]^(1x12) --> Q[x,y,z]^(1x9),
currently represented by the above matrix
julia> using CapAndHomalg
julia> LoadPackage( "GradedModulePresentationsForCAP" )
julia> ℚ = HomalgFieldOfRationalsInSingular( )
GAP: Q
julia> S = GradedRing( ℚ["x,y"] )
GAP: Q[x,y]
(weights: yet unset)
julia> Sgrmod = GradedLeftPresentations( S )
GAP: The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])
julia> InfoOfInstalledOperationsOfCategory( Sgrmod )
38 primitive operations were used to derive 329 operations for this category which algorithmically
* IsMonoidalCategory
* IsAbelianCategoryWithEnoughProjectives
and not yet algorithmically
* IsSymmetricClosedMonoidalCategory
julia> #ListPrimitivelyInstalledOperationsOfCategory( Sgrmod )
julia> M = GradedFreeLeftPresentation( 2, S, @gap([ 1, 1 ]) )
GAP: <An object in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>
julia> N = GradedFreeLeftPresentation( 1, S, @gap([ 0 ]) )
GAP: <An object in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>
julia> mat = HomalgMatrix( "[x,y]", 2, 1, S )
GAP: <A 2 x 1 matrix over a graded ring>
julia> Display( mat )
x,
y
(over a graded ring)
julia> ϕ = GradedPresentationMorphism( M, mat, N )
GAP: <A morphism in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>
julia> IsWellDefined( ϕ )
true
julia> IsMonomorphism( ϕ )
false
julia> IsEpimorphism( ϕ )
false
julia> ι = ImageEmbedding( ϕ )
GAP: <A monomorphism in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>
julia> IsMonomorphism( ι )
true
julia> IsIsomorphism( ι )
false
julia> coker_mod = CokernelObject( ϕ )
GAP: <An object in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>
julia> Display( coker_mod )
x,
y
(over a graded ring)
An object in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])
(graded, degree of generator:[ 0 ])
julia> IsZero( coker_mod )
false
julia> is_artinian = GapObj( M -> AffineDimension( M ) <= 0 );
julia> SetNameFunction( is_artinian, g"is_artinian" )
julia> C = FullSubcategoryByMembershipFunction( Sgrmod, is_artinian )
GAP: <Subcategory of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by is_artinian>
julia> CohP1 = Sgrmod / C
GAP: The Serre quotient category of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian
julia> InfoOfInstalledOperationsOfCategory( CohP1 )
21 primitive operations were used to derive 284 operations for this category which algorithmically
* IsAbelianCategory
julia> Sh = CanonicalProjection( CohP1 )
GAP: Localization functor of The Serre quotient category of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian
julia> InstallFunctor( Sh, g"Sheafification" )
julia> ψ = ApplyFunctor( Sh, ϕ )
GAP: <A morphism in The Serre quotient category of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian>
julia> IsMonomorphism( ψ )
false
julia> IsEpimorphism( ψ )
true
julia> coker_shv = CokernelObject( ψ )
GAP: <A zero object in The Serre quotient category of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian>
julia> IsZero( coker_shv )
true
julia> ϵ = ApplyFunctor( Sh, ι )
GAP: <A morphism in The Serre quotient category of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian>
julia> IsIsomorphism( ϵ )
true