using CapAndHomalg
CapAndHomalg v1.4.15
Imported OSCAR's components GAP and Singular_jll
Type: ?CapAndHomalg for more information
LoadPackage( "GradedModulePresentationsForCAP" )
ℚ = HomalgFieldOfRationalsInSingular()
GAP: Q
S = GradedRing( ℚ["x,y"] )
GAP: Q[x,y] (weights: yet unset)
Sgrmod = GradedLeftPresentations( S )
GAP: The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])
InfoOfInstalledOperationsOfCategory( Sgrmod )
40 primitive operations were used to derive 327 operations for this category which algorithmically * IsMonoidalCategory * IsAbelianCategoryWithEnoughProjectives and furthermore mathematically * IsSymmetricClosedMonoidalCategory (but not yet algorithmically)
#ListPrimitivelyInstalledOperationsOfCategory( Sgrmod )
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 ])>
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 ])>
mat = HomalgMatrix( "[x,y]", 2, 1, S )
GAP: <A 2 x 1 matrix over a graded ring>
Display( mat )
x, y (over a graded ring)
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 ])>
IsWellDefined( ϕ )
true
IsMonomorphism( ϕ )
false
IsEpimorphism( ϕ )
false
ι = ImageEmbedding( ϕ )
GAP: <A monomorphism in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>
IsMonomorphism( ι )
true
IsIsomorphism( ι )
false
coker_mod = CokernelObject( ϕ )
GAP: <An object in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>
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 ])
IsZero( coker_mod )
false
is_artinian = GapObj( M -> AffineDimension( M ) <= 0 );
SetNameFunction( is_artinian, g"is_artinian" )
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>
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
InfoOfInstalledOperationsOfCategory( CohP1 )
21 primitive operations were used to derive 280 operations for this category which algorithmically * IsAbelianCategory
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
InstallFunctor( Sh, g"Sheafification" )
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>
IsMonomorphism( ψ )
false
IsEpimorphism( ψ )
true
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>
IsZero( coker_shv )
true
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>
IsIsomorphism( ϵ )
true