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<head><title>HH^ZZ SumOfTwists -- coherent sheaf cohomology module</title>
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<div><h1>HH^ZZ SumOfTwists -- coherent sheaf cohomology module</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>HH^i(F(>=d))</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_cohomology.html" title="general cohomology functor">cohomology</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>i</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
<li><span><span>an object of class <a href="___Sum__Of__Twists.html" title="the class of all sums of twists">SumOfTwists</a></span>, <tt>F(>=d)</tt>, notation representing the sum of the twists <tt>F(n)</tt> for all <tt>n</tt> greater than or equal to <tt>d</tt>, where <tt>F</tt> is a coherent sheaf on a projective variety <tt>X</tt>.</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, M over the homogeneous coordinate ring of the variety <tt>X</tt> which agrees, at least in degrees n greater than or equal to d, with the graded module that in degree n is the<tt>i</tt>-th cohomology group of <tt>F(n)</tt>.</span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_cohomology_lp..._cm_sp__Degree_sp_eq_gt_sp..._rp.html">Degree => ...</a>, </span></li>
</ul>
</div>
</li>
</ul>
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<div class="single"><h2>Description</h2>
<div>The command computes a module over the homogeneous coordinate ring of the variety <tt>X</tt> which agrees, at least in degrees n greater than or equal to the given d, with the graded module that in degree n is the<tt>i</tt>-th cohomology group of <tt>F(n)</tt>.<p/>
To discard the part of the module M of degree less than d, use <tt>truncate(d,M)</tt>.<p/>
Use <tt>HH^i(F(>d))</tt> to request the twists strictly greater than n.<p/>
Note: use <tt>HH^i(F(*))</tt> to try to compute the whole graded module. The computation will fail if the module is not finitely generated.<p/>
As a first example we look at the cohomology of line bundles on the projective plane<table class="examples"><tr><td><pre>i1 : X = Proj(QQ[x_0..x_2])
o1 = X
o1 : ProjectiveVariety</pre>
</td></tr>
<tr><td><pre>i2 : HH^0(OO_X^1(>=0))
1
o2 = (QQ[x , x , x ])
0 1 2
o2 : QQ[x , x , x ]-module, free
0 1 2</pre>
</td></tr>
<tr><td><pre>i3 : HH^1(OO_X^1(>=0))
o3 = 0
o3 : QQ[x , x , x ]-module
0 1 2</pre>
</td></tr>
<tr><td><pre>i4 : HH^2(OO_X^1(>=-3)) -- this should change to * once implemented
o4 = cokernel {-3} | x_2 x_1 x_0 |
1
o4 : QQ[x , x , x ]-module, quotient of (QQ[x , x , x ])
0 1 2 0 1 2</pre>
</td></tr>
<tr><td><pre>i5 : TruncDual = HH^2(OO_X^1(>=-4))
o5 = cokernel {-4} | x_2 x_1 0 0 x_0 0 0 0 |
{-4} | 0 0 x_2 x_0 0 x_1 0 0 |
{-4} | 0 0 0 0 -x_2 -x_2 x_1 x_0 |
3
o5 : QQ[x , x , x ]-module, quotient of (QQ[x , x , x ])
0 1 2 0 1 2</pre>
</td></tr>
<tr><td><pre>i6 : hilbertFunction(-4, TruncDual)
o6 = 3</pre>
</td></tr>
<tr><td><pre>i7 : hilbertFunction(-3, TruncDual)
o7 = 1</pre>
</td></tr>
</table>
<p/>
As a second example we compute the H^1 cohomology module T of the Horrocks-Mumford bundle on the projective fourspace. T is an artinian module with Hilbert function (5,10,10,2):<table class="examples"><tr><td><pre>i8 : R = QQ[x_0..x_4];</pre>
</td></tr>
<tr><td><pre>i9 : a = {1,0,0,0,0}
o9 = {1, 0, 0, 0, 0}
o9 : List</pre>
</td></tr>
<tr><td><pre>i10 : b = {0,1,0,0,1}
o10 = {0, 1, 0, 0, 1}
o10 : List</pre>
</td></tr>
<tr><td><pre>i11 : c = {0,0,1,1,0}
o11 = {0, 0, 1, 1, 0}
o11 : List</pre>
</td></tr>
<tr><td><pre>i12 : M1 = matrix table(5,5, (i,j)-> x_((i+j)%5)*a_((i-j)%5))
o12 = | x_0 0 0 0 0 |
| 0 x_2 0 0 0 |
| 0 0 x_4 0 0 |
| 0 0 0 x_1 0 |
| 0 0 0 0 x_3 |
5 5
o12 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i13 : M2 = matrix table(5,5, (i,j)-> x_((i+j)%5)*b_((i-j)%5))
o13 = | 0 x_1 0 0 x_4 |
| x_1 0 x_3 0 0 |
| 0 x_3 0 x_0 0 |
| 0 0 x_0 0 x_2 |
| x_4 0 0 x_2 0 |
5 5
o13 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i14 : M3 = matrix table(5,5, (i,j)-> x_((i+j)%5)*c_((i-j)%5))
o14 = | 0 0 x_2 x_3 0 |
| 0 0 0 x_4 x_0 |
| x_2 0 0 0 x_1 |
| x_3 x_4 0 0 0 |
| 0 x_0 x_1 0 0 |
5 5
o14 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i15 : M = M1 | M2 | M3;
5 15
o15 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i16 : betti (C=res coker M)
0 1 2 3 4 5
o16 = total: 5 15 29 37 20 2
0: 5 15 10 2 . .
1: . . 4 . . .
2: . . 15 35 20 .
3: . . . . . 2
o16 : BettiTally</pre>
</td></tr>
<tr><td><pre>i17 : N = transpose submatrix(C.dd_3,{10..28},{2..36});
35 19
o17 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i18 : betti (D=res coker N)
0 1 2 3 4 5
o18 = total: 35 19 19 35 20 2
-5: 35 15 . . . .
-4: . 4 . . . .
-3: . . . . . .
-2: . . . . . .
-1: . . . . . .
0: . . 4 . . .
1: . . 15 35 20 .
2: . . . . . 2
o18 : BettiTally</pre>
</td></tr>
<tr><td><pre>i19 : Pfour = Proj(R)
o19 = Pfour
o19 : ProjectiveVariety</pre>
</td></tr>
<tr><td><pre>i20 : HorrocksMumford = sheaf(coker D.dd_3);</pre>
</td></tr>
<tr><td><pre>i21 : T = HH^1(HorrocksMumford(>=-1))
o21 = cokernel {-1} | x_4 x_2 0 0 x_0 0 0 0 x_3 0 0 0 0 0 x_1 |
{-1} | 0 -x_3 x_1 0 0 x_4 x_2 0 0 0 0 x_0 0 0 0 |
{-1} | 0 0 0 x_3 -x_2 x_0 0 x_1 0 0 0 0 0 -x_4 0 |
{-1} | 0 0 0 0 0 0 -x_4 -x_2 x_1 x_0 x_3 0 0 0 0 |
{-1} | 0 0 0 0 0 0 0 0 0 0 x_4 -x_3 x_2 x_1 x_0 |
5
o21 : R-module, quotient of R</pre>
</td></tr>
<tr><td><pre>i22 : apply(-1..2, i-> hilbertFunction(i,T))
o22 = (5, 10, 10, 2)
o22 : Sequence</pre>
</td></tr>
</table>
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<div class="single"><h2>Caveat</h2>
<div>The computation will fail if the module is not finitely generated. Also the version HH^i(F(*)) is not yet implemented.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="___H__H.html" title="general homology and cohomology functor">HH</a> -- general homology and cohomology functor</span></li>
<li><span><a href="___H__H^__Z__Z_sp__Coherent__Sheaf.html" title="cohomology of a coherent sheaf on a projective variety">HH^ZZ CoherentSheaf</a> -- cohomology of a coherent sheaf on a projective variety</span></li>
<li><span><a href="___H__H^__Z__Z_sp__Module.html" title="local cohomology of a module">HH^ZZ Module</a> -- local cohomology of a module</span></li>
</ul>
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