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<head><title>pureCharFree -- first betti number of specific exact complex</title>
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<div><h1>pureCharFree -- first betti number of specific exact complex</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>pureCharFree</tt></div>
</dd></dl>
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<li><div class="single">Inputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a strictly increasing sequence of degrees</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, The zero-th betti number of the corresponding pure resolution construction</span></li>
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<div class="single"><h2>Description</h2>
<div><tt>pureCharFree</tt> corresponds to the construction in math.AC/0712.1843v2, "Betti Numbers of Graded Modules and Cohomology of Vector Bundles", Section 5.<table class="examples"><tr><td><pre>i1 : L = {0,2,3,9}
o1 = {0, 2, 3, 9}
o1 : List</pre>
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<tr><td><pre>i2 : B = pureBettiDiagram L
0 1 2 3
o2 = total: 7 27 21 1
0: 7 . . .
1: . 27 21 .
2: . . . .
3: . . . .
4: . . . .
5: . . . .
6: . . . 1
o2 : BettiTally</pre>
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<tr><td><pre>i3 : pureCharFree L
o3 = 56</pre>
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<tr><td><pre>i4 : L1 = {0,3,4,6}
o4 = {0, 3, 4, 6}
o4 : List</pre>
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<tr><td><pre>i5 : B1 = pureBettiDiagram L1
0 1 2 3
o5 = total: 1 8 9 2
0: 1 . . .
1: . . . .
2: . 8 9 .
3: . . . 2
o5 : BettiTally</pre>
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<tr><td><pre>i6 : pureCharFree L1
o6 = 5</pre>
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Thus, for large enough multiples m, m*B occurs as the Betti diagram of a module from the pureCharFree construction<p/>
However, we can find B itself as the Betti diagram of a module:<table class="examples"><tr><td><pre>i7 : betti res randomSocleModule(L,1)
0 1 2 3
o7 = total: 7 27 21 1
0: 7 . . .
1: . 27 21 .
2: . . . .
3: . . . .
4: . . . .
5: . . . .
6: . . . 1
o7 : BettiTally</pre>
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<tr><td><pre>i8 : betti res randomModule(L,1)
0 1 2 3
o8 = total: 7 27 35 15
0: 7 . . .
1: . 27 11 .
2: . . 24 15
o8 : BettiTally</pre>
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<tr><td><pre>i9 : betti res randomModule({0,6,7,9},1)
0 1 2 3
o9 = total: 1 21 27 7
0: 1 . . .
1: . . . .
2: . . . .
3: . . . .
4: . . . .
5: . 21 27 .
6: . . . 7
o9 : BettiTally</pre>
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<tr><td><pre>i10 : betti res randomSocleModule(L1,1)
0 1 2 3
o10 = total: 1 8 9 2
0: 1 . . .
1: . . . .
2: . 8 9 .
3: . . . 2
o10 : BettiTally</pre>
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<tr><td><pre>i11 : betti res randomModule(L1,1)
0 1 2 3
o11 = total: 1 8 9 2
0: 1 . . .
1: . . . .
2: . 8 9 .
3: . . . 2
o11 : BettiTally</pre>
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<tr><td><pre>i12 : betti res randomModule({0,2,3,6},1)
0 1 2 3
o12 = total: 2 9 10 3
0: 2 . . .
1: . 9 7 .
2: . . 3 3
o12 : BettiTally</pre>
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<tr><td><pre>i13 : betti res randomSocleModule({0,2,3,6},1)
0 1 2 3
o13 = total: 2 9 8 1
0: 2 . . .
1: . 9 8 .
2: . . . .
3: . . . 1
o13 : BettiTally</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_pure__All.html" title="Vector of first betti number of our three specific exact complexes">pureAll</a> -- Vector of first betti number of our three specific exact complexes</span></li>
<li><span><a href="_pure__Weyman.html" title="first betti number of specific exact complex">pureWeyman</a> -- first betti number of specific exact complex</span></li>
<li><span><a href="_pure__Two__Invariant.html" title="first betti number of specific exact complex">pureTwoInvariant</a> -- first betti number of specific exact complex</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>pureCharFree</tt> :</h2>
<ul><li>pureCharFree(List)</li>
</ul>
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