‣ CoefficientsRing ( AffAlg_k ) | ( attribute ) |
The input is a category of finitely presented associative commutative unital \(k\)-algebras. The output is the underlying commutative ring \(k\) of coefficients.
‣ DefiningSextupleOfAffineAlgebra ( affine_algebra ) | ( attribute ) |
‣ MatrixOfImages ( affine_algebra_morphism ) | ( attribute ) |
‣ Dimension ( affine_algebra ) | ( attribute ) |
The input is a finitely presented associative commutative unital algebra. The output is the corresponding ambient (free) polynomial algebra.
‣ AmbientAlgebra ( affine_algebra ) | ( attribute ) |
The input is a finitely presented associative commutative unital algebra. The output is the corresponding ambient (free) polynomial algebra.
‣ CategoryOfAffineAlgebras ( k ) | ( attribute ) |
The input is a commutative ring k. The output is the category of finitely presented associative commutative unital algebras over \(k\).
gap> LoadPackage( "ZariskiFrames", false ); true gap> zz := HomalgRingOfIntegersInSingular( ); Z gap> AffAlg_Z := CategoryOfAffineAlgebras( zz ); CategoryOfAffineAlgebras( Z ) gap> Display( AffAlg_Z ); A CAP category with name CategoryOfAffineAlgebras( Z ): 27 primitive operations were used to derive 154 operations for this category \ which algorithmically * IsFiniteCocompleteCategory * IsSymmetricMonoidalCategory * IsCodistributiveCategory and furthermore mathematically * IsSymmetricMonoidalCategoryStructureGivenByCoproduct gap> AffSch_Z := Opposite( AffAlg_Z : only_primitive_operations := true ); Opposite( CategoryOfAffineAlgebras( Z ) ) gap> Display( AffSch_Z ); A CAP category with name Opposite( CategoryOfAffineAlgebras( Z ) ): 27 primitive operations were used to derive 160 operations for this category \ which algorithmically * IsFiniteCompleteCategory * IsSymmetricMonoidalCategory * IsDistributiveCategory and furthermore mathematically * IsSymmetricMonoidalCategoryStructureGivenByDirectProduct gap> iota := UniversalMorphismFromInitialObject( TerminalObject( AffAlg_Z ) ); <A morphism in CategoryOfAffineAlgebras( Z )> gap> Display( iota ); Z/( 1 ) ^ | [ ] | | Z/( 0 ) gap> IsWellDefined( iota ); true gap> terminal := Source( iota ); <An object in CategoryOfAffineAlgebras( Z )> gap> Dimension( terminal ); 1 gap> initial := Target( iota ); <An object in CategoryOfAffineAlgebras( Z )> gap> Dimension( initial ); -1 gap> uniq := UniversalMorphismIntoTerminalObject( InitialObject( AffAlg_Z ) ); <A morphism in CategoryOfAffineAlgebras( Z )> gap> Display( uniq ); Z/( 1 ) ^ | [ ] | | Z/( 0 ) gap> IsWellDefined( uniq ); true gap> iota = uniq; true gap> W := ( zz["e"] / "e^2-e" ) / AffAlg_Z; <An object in CategoryOfAffineAlgebras( Z )> gap> Display( W ); Z[e]/( e^2-e ) gap> Dimension( W ); 1 gap> IsWellDefined( W ); true gap> S := zz["x,y,z"] / AffAlg_Z; <An object in CategoryOfAffineAlgebras( Z )> gap> Display( S ); Z[x,y,z]/( 0 ) gap> Dimension( S ); 4 gap> IsWellDefined( S ); true gap> T := zz["u"] / AffAlg_Z; <An object in CategoryOfAffineAlgebras( Z )> gap> Display( T ); Z[u]/( 0 ) gap> Dimension( T ); 2 gap> IsWellDefined( T ); true gap> ExportVariables( ObjectDatum( T )[1] ); [ u ] gap> phi := MorphismConstructor( S, [ u, u^2, u^3 ], T ); <A morphism in CategoryOfAffineAlgebras( Z )> gap> Display( phi ); Z[u]/( 0 ) ^ | [ |[ u ]|, |[ u^2 ]|, |[ u^3 ]| ] | | Z[x,y,z]/( 0 ) gap> IsWellDefined( phi ); true gap> coimage := CoimageObject( phi ); <An object in CategoryOfAffineAlgebras( Z )> gap> Display( coimage ); Z[x,y,z]/( y^2-x*z, x*y-z, x^2-y ) gap> Dimension( coimage ); 2 gap> IsWellDefined( coimage ); true gap> prj := CoimageProjection( phi ); <A morphism in CategoryOfAffineAlgebras( Z )> gap> Display( prj ); Z[x,y,z]/( y^2-x*z, x*y-z, x^2-y ) ^ | [ |[ x ]|, |[ y ]|, |[ z ]| ] | | Z[x,y,z]/( 0 ) gap> IsWellDefined( prj ); true gap> IsEpimorphism( prj ); true gap> IsMonomorphism( prj ); false gap> ast := AstrictionToCoimage( phi ); <A morphism in CategoryOfAffineAlgebras( Z )> gap> Display( ast ); Z[u]/( 0 ) ^ | [ |[ u ]|, |[ u^2 ]|, |[ u^3 ]| ] | | Z[x,y,z]/( y^2-x*z, x*y-z, x^2-y ) gap> IsWellDefined( ast ); true gap> IsIsomorphism( ast ); true gap> inv := InverseForMorphisms( ast ); <An isomorphism in CategoryOfAffineAlgebras( Z )> gap> Display( inv ); Z[x,y,z]/( y^2-x*z, x*y-z, x^2-y ) ^ | [ |[ x ]| ] | | Z[u]/( 0 ) gap> IsWellDefined( inv ); true gap> IsOne( PreCompose( ast, inv ) ); true gap> IsOne( PreCompose( inv, ast ) ); true gap> chi := MorphismConstructor( S, [ u^2, u^3, u^4 ], T ); <A morphism in CategoryOfAffineAlgebras( Z )> gap> Display( chi ); Z[u]/( 0 ) ^ | [ |[ u^2 ]|, |[ u^3 ]|, |[ u^4 ]| ] | | Z[x,y,z]/( 0 ) gap> IsWellDefined( chi ); true gap> coeq := Coequalizer( phi, chi ); <An object in CategoryOfAffineAlgebras( Z )> gap> Display( coeq ); Z[u]/( -u^2+u, -u^3+u^2, -u^4+u^3 ) gap> Dimension( coeq ); 1 gap> Display( coeq ); Z[u]/( u^2-u ) gap> pi := ProjectionOntoCoequalizer( [ phi, chi ] ); <An epimorphism in CategoryOfAffineAlgebras( Z )> gap> IsOne( UniversalMorphismFromCoequalizer( [ phi, chi ], pi ) ); true gap> P := DirectProduct( coimage, coeq ); <An object in CategoryOfAffineAlgebras( Z )> gap> Display( P ); Z[c1,c2,c3,c4,c5,c6]/( c2^2-c1*c3, c1*c2-c3, c1^2-c2, -c5^2+c5, -c5^3+c5^2, \ -c5^4+c5^3, c4^2-c4, c6^2-c6, c4*c6, c1*c4-c1, c2*c4-c2, c3*c4-c3, c5*c6-c5, -c4-c6+1 ) gap> C := Coproduct( coimage, coeq ); <An object in CategoryOfAffineAlgebras( Z )> gap> Display( C ); Z[c1,c2,c3,c4]/( c2^2-c1*c3, c1*c2-c3, c1^2-c2, -c4^2+c4, -c4^3+c4^2, \ -c4^4+c4^3 ) gap> C = TensorProduct( coimage, coeq ); true
‣ IsCategoryOfAffineAlgebras ( category ) | ( filter ) |
Returns: true
or false
The GAP category of categories of finitely presented associative commutative unital algebras.
‣ IsObjectInCategoryOfAffineAlgebras ( object ) | ( filter ) |
Returns: true
or false
The GAP category of finitely presented associative commutative unital algebras.
‣ IsMorphismInCategoryOfAffineAlgebras ( morphism ) | ( filter ) |
Returns: true
or false
The GAP category of morphisms of finitely presented associative commutative unital algebras.
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