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### 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.

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 ].

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)

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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