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<head><title>betti(GradedModule) -- display of degrees in a graded module</title>
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<div><h1>betti(GradedModule) -- display of degrees in a graded 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>betti C</tt></div>
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<li><span>Function: <a href="_betti.html" title="display degrees">betti</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>C</tt>, <span>a <a href="___Graded__Module.html">graded module</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Betti__Tally.html">Betti tally</a></span>, a diagram showing the degrees of the generators of the modules in <tt>C</tt></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>Weights => </tt><span><span>a <a href="___List.html">list</a></span>, <span>default value null</span>, a list of integers whose dot product with the multidegree of a basis element is enumerated in the display returned. The default is the heft vector of the ring. See <a href="_heft_spvectors.html" title="">heft vectors</a>.</span></span></li>
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<div class="single"><h2>Description</h2>
<div><p>The diagram can be used to determine the degrees of the entries in the matrices of the differentials in a chain complex (which is a type of graded module) provided they are homogeneous maps of degree 0.</p>
<p>Here is a sample diagram.</p>
<table class="examples"><tr><td><pre>i1 : R = ZZ/101[a..h]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : p = genericMatrix(R,a,2,4)
o2 = | a c e g |
| b d f h |
2 4
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : q = generators gb p
o3 = | g e c a 0 0 0 0 0 0 |
| h f d b fg-eh dg-ch bg-ah de-cf be-af bc-ad |
2 10
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : C = resolution cokernel leadTerm q
2 10 14 7 1
o4 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o4 : ChainComplex</pre>
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<tr><td><pre>i5 : betti C
0 1 2 3 4
o5 = total: 2 10 14 7 1
0: 2 4 6 4 1
1: . 6 8 3 .
o5 : BettiTally</pre>
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<p>Column <tt>j</tt> of the top row of the diagram gives the rank of the free module <tt>C_j</tt>. (Columns are numbered from 0.) The entry in column <tt>j</tt> in the row labelled <tt>i</tt> is the number of basis elements of (weighted) degree <tt>i+j</tt> in the free module<tt> C_j</tt>. When the chain complex is the resolution of a module the entries are the total and the graded Betti numbers of the module.</p>
<p>If the numbers are needed in a program, then they are accessible, because the value returned is <span>a <a href="___Betti__Tally.html">Betti tally</a></span>, and the diagram you see on the screen is just the way it prints out.</p>
<p>The heft vector is used, by default, as the weight vector for weighting the components of the degree vectors of basis elements.</p>
<table class="examples"><tr><td><pre>i6 : R = QQ[a,b,c,Degrees=>{-1,-2,-3}];</pre>
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<tr><td><pre>i7 : heft R
o7 = {-1}
o7 : List</pre>
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<tr><td><pre>i8 : betti res coker vars R
0 1 2 3
o8 = total: 1 3 3 1
0: 1 1 . .
1: . 1 1 .
2: . 1 1 .
3: . . 1 1
o8 : BettiTally</pre>
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<tr><td><pre>i9 : betti(oo, Weights => {1})
0 1 2 3
o9 = total: 1 3 3 1
-9: . . . 1
-8: . . . .
-7: . . 1 .
-6: . . 1 .
-5: . . 1 .
-4: . 1 . .
-3: . 1 . .
-2: . 1 . .
-1: . . . .
0: 1 . . .
o9 : BettiTally</pre>
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<tr><td><pre>i10 : R = QQ[a,b,c,d,Degrees=>{{1,0},{2,1},{0,1},{-2,1}}];</pre>
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<tr><td><pre>i11 : heft R
o11 = {1, 3}
o11 : List</pre>
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<tr><td><pre>i12 : b = betti res coker vars R
0 1 2 3 4
o12 = total: 1 4 6 4 1
0: 1 2 1 . .
1: . . . . .
2: . 1 2 1 .
3: . . . . .
4: . 1 2 1 .
5: . . . . .
6: . . 1 2 1
o12 : BettiTally</pre>
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<tr><td><pre>i13 : betti(b, Weights => {1,0})
0 1 2 3 4
o13 = total: 1 4 6 4 1
-4: . . 1 1 .
-3: . 1 1 1 1
-2: . . 1 1 .
-1: . 1 1 . .
0: 1 1 1 1 .
1: . 1 1 . .
o13 : BettiTally</pre>
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<tr><td><pre>i14 : betti(b, Weights => {0,1})
0 1 2 3 4
o14 = total: 1 4 6 4 1
-1: . 1 3 3 1
0: 1 3 3 1 .
o14 : BettiTally</pre>
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