6 The Tate Resolution
  
  
  6.1 The Tate Resolution: Operations and Functions
  
  6.1-1 TateResolution
  
  TateResolution( M, degree_lowest, degree_highest )  operation
  Returns:  a homalg cocomplex
  
  Compute the Tate resolution of the sheaf M.
  
    Example  
    gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0..x3";;
    gap> S := GradedRing( R );;
    gap> A := KoszulDualRing( S, "e0..e3" );;
  
  
  In the following we construct the different exterior powers of the cotangent
  bundle  shifted  by  1. Observe how a single 1 travels along the diagnoal in
  the window [ -3 .. 0 ] x [ 0 .. 3 ].
  First we start with the structure sheaf with its Tate resolution:
  
    Example  
    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
  
  
  The Castelnuovo-Mumford regularity of the underlying module is distinguished
  among  the  list of twists by the character 'V' pointing to it. It is not an
  invariant of the sheaf (see the next diagram).
  The residue class field (i.e. S modulo the maximal homogeneous ideal):
  
    Example  
    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>
  
  
  Another way of constructing the structure sheaf:
  
    Example  
    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
  
  
  The cotangent bundle:
  
    Example  
    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
  
  
  the cotangent bundle shifted by 1 with its Tate resolution:
  
    Example  
    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
  
  
  The  second  power  U^2  of  the shifted cotangent bundle U=U^1 and its Tate
  resolution:
  
    Example  
    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
  
  
  The  third  power  U^3  of  the  shifted cotangent bundle U=U^1 and its Tate
  resolution:
  
    Example  
    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
  
  
  Another way to construct U^2=U^(3-1):
  
    Example  
    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