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<head><title>cohomologyTable -- dimensions of cohomology groups</title>
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<div><h1>cohomologyTable -- dimensions of cohomology groups</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>cohomologyTable(F,l,h) or cohomologyTable(m,E,l,h)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>a <a href="../../Macaulay2Doc/html/___Coherent__Sheaf.html">coherent sheaf</a></span>, a coherent sheaf on a projective scheme</span></li>
<li><span><tt>l</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, lower cohomological degree</span></li>
<li><span><tt>h</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, upper bound on the cohomological degree</span></li>
<li><span><tt>m</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, a presentation matrix for a module</span></li>
<li><span><tt>E</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span>, exterior algebra</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><a href="../../BoijSoederberg/html/___Cohomology__Tally.html" title="cohomology table">CohomologyTally</a> dimensions of cohomology groups</span></li>
</ul>
</div>
</li>
</ul>
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<div class="single"><h2>Description</h2>
<div> This function takes as input a coherent sheaf <tt>F</tt>, two integers <tt>l</tt> and <tt>h</tt>, and prints the dimension <tt>dim HH^j F(i-j)</tt> for <tt>h>=i>=l</tt>. As a simple example, we compute the dimensions of cohomology groups of the projective plane.<table class="examples"><tr><td><pre>i1 : S = ZZ/32003[x_0..x_2]; </pre>
</td></tr>
<tr><td><pre>i2 : PP2 = Proj S; </pre>
</td></tr>
<tr><td><pre>i3 : F =sheaf S^1
1
o3 = OO
PP2
o3 : coherent sheaf on PP2, free</pre>
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<tr><td><pre>i4 : cohomologyTable(F,-10,5)
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
o4 = 2: 55 45 36 28 21 15 10 6 3 1 . . . . . .
1: . . . . . . . . . . . . . . . .
0: . . . . . . . . . . 1 3 6 10 15 21
o4 : CohomologyTally</pre>
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There is also a built-in sheaf cohomology function <a href="../../Macaulay2Doc/html/___H__H^__Z__Z_sp__Coherent__Sheaf.html">HH</a> in Macaulay2. However, these algorithms are much slower than <tt>cohomologyTable</tt>.<p/>
Alternatively, this function takes as input a presentation matrix <tt>m</tt> of a finitely generated graded <tt>S</tt>-module <tt>M</tt>and an exterior algebra <tt>E</tt>with the same number of variables. In this form, the function is equivalent to the fucntion <tt>sheafCohomology</tt> in <a href="http://www.math.uiuc.edu/Macaulay2/Book/">Sheaf Algorithms Using Exterior Algebra</a>.<table class="examples"><tr><td><pre>i5 : S = ZZ/32003[x_0..x_2]; </pre>
</td></tr>
<tr><td><pre>i6 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true];</pre>
</td></tr>
<tr><td><pre>i7 : m = koszul (3, vars S);
3 1
o7 : Matrix S <--- S</pre>
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<tr><td><pre>i8 : regularity coker m
o8 = 2</pre>
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<tr><td><pre>i9 : betti tateResolution(m,E,-6,2)
0 1 2 3 4 5 6 7 8 9 10
o9 = total: 15 8 3 1 3 8 15 24 35 48 63
-4: 15 8 3 . . . . . . . .
-3: . . . 1 . . . . . . .
-2: . . . . 3 8 15 24 35 48 63
o9 : BettiTally</pre>
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<tr><td><pre>i10 : cohomologyTable(m,E,-6,2)
-6 -5 -4 -3 -2 -1 0 1 2 3 4
o10 = 2: 63 48 35 24 15 8 3 . . . .
1: . . . . . . . 1 . . .
0: . . . . . . . . 3 8 15
o10 : CohomologyTally</pre>
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<p/>
As the third example, we compute the dimensions of cohomology groups of the structure sheaf of an irregular elliptic surface.<table class="examples"><tr><td><pre>i11 : S = ZZ/32003[x_0..x_4]; </pre>
</td></tr>
<tr><td><pre>i12 : X = Proj S; </pre>
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<tr><td><pre>i13 : ff = res coker map(S^{1:0},S^{3:-1,2:-2},{{x_0..x_2,x_3^2,x_4^2}}); </pre>
</td></tr>
<tr><td><pre>i14 : alpha = map(S^{1:-2},target ff.dd_3,{{1,4:0,x_0,2:0,x_1,0}})*ff.dd_3;
1 10
o14 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i15 : beta = ff.dd_4//syz alpha;
18 5
o15 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i16 : K = syz syz alpha|beta;
18 21
o16 : Matrix S <--- S</pre>
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<tr><td><pre>i17 : fK = res prune coker K;</pre>
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<tr><td><pre>i18 : s = random(target fK.dd_1,S^{1:-4,3:-5});
13 4
o18 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i19 : ftphi = res prune coker transpose (fK.dd_1|s);</pre>
</td></tr>
<tr><td><pre>i20 : I = ideal ftphi.dd_2;
o20 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i21 : F = sheaf S^1/I; </pre>
</td></tr>
<tr><td><pre>i22 : cohomologyTable(F,-2,6)
-2 -1 0 1 2 3 4 5 6 7 8
o22 = 2: 123 75 39 15 3 . . . . . .
1: . . . 1 2 1 . . . . .
0: . . 1 5 16 39 75 123 183 255 339
o22 : CohomologyTally</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_sym__Ext.html" title="the first differential of the complex R(M)">symExt</a> -- the first differential of the complex R(M)</span></li>
<li><span><a href="_tate__Resolution.html" title="finite piece of the Tate resolution">tateResolution</a> -- finite piece of the Tate resolution</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>cohomologyTable</tt> :</h2>
<ul><li>cohomologyTable(CoherentSheaf,ZZ,ZZ)</li>
<li>cohomologyTable(Matrix,PolynomialRing,ZZ,ZZ)</li>
</ul>
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