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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418384## <#GAPDoc Label="TateResolution:example1"> ## <Example><![CDATA[ ## gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0..x3";; ## gap> S := GradedRing( R );; ## gap> A := KoszulDualRing( S, "e0..e3" );; ## ]]></Example> ## <#/GAPDoc> LoadPackage( "GradedModules" ); R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0..x3";; S := GradedRing( R );; A := KoszulDualRing( S );; ## <#GAPDoc Label="TateResolution:example2"> ## <Example><![CDATA[ ## gap> O := S^0; ## <The graded free left module of rank 1 on a free generator> ## gap> T := TateResolution( O, -5, 5 ); ## <An acyclic cocomplex containing ## 10 morphisms of graded left modules at degrees [ -5 .. 5 ]> ## gap> betti := BettiTable( T ); ## <A Betti diagram of <An acyclic cocomplex containing ## 10 morphisms of graded left modules at degrees [ -5 .. 5 ]>> ## gap> Display( betti ); ## total: 35 20 10 4 1 1 4 10 20 35 56 ? ? ? ## ----------|---|---|---|---|---|---|---|---|---|---|---|---|---| ## 3: 35 20 10 4 1 . . . . . . 0 0 0 ## 2: * . . . . . . . . . . . 0 0 ## 1: * * . . . . . . . . . . . 0 ## 0: * * * . . . . . 1 4 10 20 35 56 ## ----------|---|---|---|---|---|---|---|---S---|---|---|---|---| ## twist: -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 ## --------------------------------------------------------------- ## Euler: -35 -20 -10 -4 -1 0 0 0 1 4 10 20 35 56 ## ]]></Example> ## <#/GAPDoc> O := S^0; T := TateResolution( O, -5, 5 ); betti := BettiTable( T ); Display( betti ); ## <#GAPDoc Label="TateResolution:example3"> ## <Example><![CDATA[ ## gap> k := HomalgMatrix( Indeterminates( S ), Length( Indeterminates( S ) ), 1, S ); ## <A 4 x 1 matrix over a graded ring> ## gap> k := LeftPresentationWithDegrees( k ); ## <A graded cyclic left module presented by 4 relations for a cyclic generator> ## ]]></Example> ## <#/GAPDoc> k := HomalgMatrix( Indeterminates( S ), Length( Indeterminates( S ) ), 1, S ); k := LeftPresentationWithDegrees( k ); ## <#GAPDoc Label="TateResolution:example4"> ## <Example><![CDATA[ ## gap> U0 := SyzygiesObject( 1, k ); ## <A graded torsion-free left module presented by yet unknown relations for 4 ge\ ## nerators> ## gap> T0 := TateResolution( U0, -5, 5 ); ## <An acyclic cocomplex containing ## 10 morphisms of graded left modules at degrees [ -5 .. 5 ]> ## gap> betti0 := BettiTable( T0 ); ## <A Betti diagram of <An acyclic cocomplex containing ## 10 morphisms of graded left modules at degrees [ -5 .. 5 ]>> ## gap> Display( betti0 ); ## total: 35 20 10 4 1 1 4 10 20 35 56 ? ? ? ## ----------|---|---|---|---|---|---|---|---|---|---|---|---|---| ## 3: 35 20 10 4 1 . . . . . . 0 0 0 ## 2: * . . . . . . . . . . . 0 0 ## 1: * * . . . . . . . . . . . 0 ## 0: * * * . . . . . 1 4 10 20 35 56 ## ----------|---|---|---|---|---|---|---|---S---|---|---|---|---| ## twist: -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 ## --------------------------------------------------------------- ## Euler: -35 -20 -10 -4 -1 0 0 0 1 4 10 20 35 56 ## ]]></Example> ## <#/GAPDoc> U0 := SyzygiesObject( 1, k ); T0 := TateResolution( U0, -5, 5 ); betti0 := BettiTable( T0 ); Display( betti0 ); ## <#GAPDoc Label="TateResolution:example5"> ## <Example><![CDATA[ ## gap> cotangent := SyzygiesObject( 2, k ); ## <A graded torsion-free left module presented by yet unknown relations for 6 ge\ ## nerators> ## gap> IsFree( UnderlyingModule( cotangent ) ); ## false ## gap> Rank( cotangent ); ## 3 ## gap> cotangent; ## <A graded reflexive non-projective rank 3 left module presented by 4 relations\ ## for 6 generators> ## gap> ProjectiveDimension( UnderlyingModule( cotangent ) ); ## 2 ## ]]></Example> ## <#/GAPDoc> cotangent := SyzygiesObject( 2, k ); IsFree( UnderlyingModule( cotangent ) ); Rank( cotangent ); cotangent; ProjectiveDimension( UnderlyingModule( cotangent ) ); ## <#GAPDoc Label="TateResolution:example6"> ## <Example><![CDATA[ ## gap> U1 := cotangent * S^1; ## <A graded non-torsion left module presented by 4 relations for 6 generators> ## gap> T1 := TateResolution( U1, -5, 5 ); ## <An acyclic cocomplex containing ## 10 morphisms of graded left modules at degrees [ -5 .. 5 ]> ## gap> betti1 := BettiTable( T1 ); ## <A Betti diagram of <An acyclic cocomplex containing ## 10 morphisms of graded left modules at degrees [ -5 .. 5 ]>> ## gap> Display( betti1 ); ## total: 120 70 36 15 4 1 6 20 45 84 140 ? ? ? ## -----------|----|----|----|----|----|----|----|----|----|----|----|----|----| ## 3: 120 70 36 15 4 . . . . . . 0 0 0 ## 2: * . . . . . . . . . . . 0 0 ## 1: * * . . . . . 1 . . . . . 0 ## 0: * * * . . . . . . 6 20 45 84 140 ## -----------|----|----|----|----|----|----|----|----|----S----|----|----|----| ## twist: -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 ## ----------------------------------------------------------------------------- ## Euler: -120 -70 -36 -15 -4 0 0 -1 0 6 20 45 84 140 ## ]]></Example> ## <#/GAPDoc> U1 := cotangent * S^1; T1 := TateResolution( U1, -5, 5 ); betti1 := BettiTable( T1 ); Display( betti1 ); ## <#GAPDoc Label="TateResolution:example7"> ## <Example><![CDATA[ ## gap> U2 := SyzygiesObject( 3, k ) * S^2; ## <A graded rank 3 left module presented by 1 relation for 4 generators> ## gap> T2 := TateResolution( U2, -5, 5 ); ## <An acyclic cocomplex containing ## 10 morphisms of graded left modules at degrees [ -5 .. 5 ]> ## gap> betti2 := BettiTable( T2 ); ## <A Betti diagram of <An acyclic cocomplex containing ## 10 morphisms of graded left modules at degrees [ -5 .. 5 ]>> ## gap> Display( betti2 ); ## total: 140 84 45 20 6 1 4 15 36 70 120 ? ? ? ## -----------|----|----|----|----|----|----|----|----|----|----|----|----|----| ## 3: 140 84 45 20 6 . . . . . . 0 0 0 ## 2: * . . . . . 1 . . . . . 0 0 ## 1: * * . . . . . . . . . . . 0 ## 0: * * * . . . . . . 4 15 36 70 120 ## -----------|----|----|----|----|----|----|----|----|----S----|----|----|----| ## twist: -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 ## ----------------------------------------------------------------------------- ## Euler: -140 -84 -45 -20 -6 0 1 0 0 4 15 36 70 120 ## ]]></Example> ## <#/GAPDoc> U2 := SyzygiesObject( 3, k ) * S^2; T2 := TateResolution( U2, -5, 5 ); betti2 := BettiTable( T2 ); Display( betti2 ); ## <#GAPDoc Label="TateResolution:example8"> ## <Example><![CDATA[ ## gap> U3 := SyzygiesObject( 4, k ) * S^3; ## <A graded free left module of rank 1 on a free generator> ## gap> Display( U3 ); ## Q[x0,x1,x2,x3]^(1 x 1) ## ## (graded, degree of generator: 1) ## gap> T3 := TateResolution( U3, -5, 5 ); ## <An acyclic cocomplex containing ## 10 morphisms of graded left modules at degrees [ -5 .. 5 ]> ## gap> betti3 := BettiTable( T3 ); ## <A Betti diagram of <An acyclic cocomplex containing ## 10 morphisms of graded left modules at degrees [ -5 .. 5 ]>> ## gap> Display( betti3 ); ## total: 56 35 20 10 4 1 1 4 10 20 35 ? ? ? ## ----------|---|---|---|---|---|---|---|---|---|---|---|---|---| ## 3: 56 35 20 10 4 1 . . . . . 0 0 0 ## 2: * . . . . . . . . . . . 0 0 ## 1: * * . . . . . . . . . . . 0 ## 0: * * * . . . . . . 1 4 10 20 35 ## ----------|---|---|---|---|---|---|---|---|---S---|---|---|---| ## twist: -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 ## --------------------------------------------------------------- ## Euler: -56 -35 -20 -10 -4 -1 0 0 0 1 4 10 20 35 ## ]]></Example> ## <#/GAPDoc> U3 := SyzygiesObject( 4, k ) * S^3; Display( U3 ); T3 := TateResolution( U3, -5, 5 ); betti3 := BettiTable( T3 ); Display( betti3 ); ## <#GAPDoc Label="TateResolution:example9"> ## <Example><![CDATA[ ## gap> u2 := GradedHom( U1, S^(-1) ); ## <A graded torsion-free right module on 4 generators satisfying yet unknown rel\ ## ations> ## gap> t2 := TateResolution( u2, -5, 5 ); ## <An acyclic cocomplex containing ## 10 morphisms of graded right modules at degrees [ -5 .. 5 ]> ## gap> BettiTable( t2 ); ## <A Betti diagram of <An acyclic cocomplex containing ## 10 morphisms of graded right modules at degrees [ -5 .. 5 ]>> ## gap> Display( last ); ## total: 140 84 45 20 6 1 4 15 36 70 120 ? ? ? ## -----------|----|----|----|----|----|----|----|----|----|----|----|----|----| ## 3: 140 84 45 20 6 . . . . . . 0 0 0 ## 2: * . . . . . 1 . . . . . 0 0 ## 1: * * . . . . . . . . . . . 0 ## 0: * * * . . . . . . 4 15 36 70 120 ## -----------|----|----|----|----|----|----|----|----|----S----|----|----|----| ## twist: -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 ## ----------------------------------------------------------------------------- ## Euler: -140 -84 -45 -20 -6 0 1 0 0 4 15 36 70 120 ## ]]></Example> ## <#/GAPDoc> u2 := GradedHom( U1, S^(-1) ); t2 := TateResolution( u2, -5, 5 ); b2 := BettiTable( t2 ); Display( b2 );