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## <#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", false );
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 );
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