GAP (GradedModules) - Chapter 4: The Tate Resolution

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4 The Tate Resolution
4.1 The Tate Resolution: Operations and Functions
4.1-1 TateResolution

### 4 The Tate Resolution

#### 4.1 The Tate Resolution: Operations and Functions

##### 4.1-1 TateResolution

`‣ TateResolution`( M, degree_lowest, degree_highest ) | ( operation )

Returns: a homalg cocomplex

Compute the Tate resolution of the sheaf M.

    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:

    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):

    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:

    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:

    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:

    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:

    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:

    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 \(U^2=U^(3-1)\):

    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

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