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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 := BettiDiagram( 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 := BettiDiagram( 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 := BettiDiagram( 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 := BettiDiagram( 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 := BettiDiagram( 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> BettiDiagram( 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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