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<head><title>decompose(BettiTally) -- write a Betti diagram as a positive combination of pure integral diagrams</title>
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<div><h1>decompose(BettiTally) -- write a Betti diagram as a positive combination of pure integral diagrams</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>decompose B</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="../../Macaulay2Doc/html/_minimal__Primes.html" title="minimal associated primes of an ideal">decompose</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>B</tt>, <span>a <a href="../../Macaulay2Doc/html/___Betti__Tally.html">Betti tally</a></span>, not necessarily Cohen-Macaulay</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Expression.html">expression</a></span>, a positive combination of pure integral Betti diagrams</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>This applies the algorithm implied by the Boij-Soederberg conjecture, and also works even if the diagram does not corresponds to a Cohen-Macaulay module.<table class="examples"><tr><td><pre>i1 : R = ZZ/103[a,b,c]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : I = ideal"a3,abc,b4,c4,b2c2"
3 4 4 2 2
o2 = ideal (a , a*b*c, b , c , b c )
o2 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i3 : B = betti res I
0 1 2 3
o3 = total: 1 5 8 4
0: 1 . . .
1: . . . .
2: . 2 . .
3: . 3 2 .
4: . . 4 2
5: . . 2 2
o3 : BettiTally</pre>
</td></tr>
<tr><td><pre>i4 : C = decompose B
1 / 0 1 2 3\ 1 / 0 1 2 3\ 3 / 0 1 2 3\ 11 / 0 1 2 3\ 1 / 0 1 2 3\
o4 = (--)|total: 8 35 42 15| + (--)|total: 2 7 14 9| + (--)|total: 1 7 14 8| + (--)|total: 1 6 8 3| + (--)|total: 3 14 32 21|
21 | 0: 8 . . .| 21 | 0: 2 . . .| 28 | 0: 1 . . .| 48 | 0: 1 . . .| 16 | 0: 3 . . .|
| 1: . . . .| | 1: . . . .| | 1: . . . .| | 1: . . . .| | 1: . . . .|
| 2: . 35 . .| | 2: . 7 . .| | 2: . . . .| | 2: . . . .| | 2: . . . .|
| 3: . . 42 .| | 3: . . . .| | 3: . 7 . .| | 3: . 6 . .| | 3: . 14 . .|
\ 4: . . . 15/ \ 4: . . 14 9/ \ 4: . . 14 8/ | 4: . . 8 .| | 4: . . . .|
\ 5: . . . 3/ \ 5: . . 32 21/
o4 : Expression of class Sum</pre>
</td></tr>
</table>
Check that this really does sum to B:<table class="examples"><tr><td><pre>i5 : value C
0 1 2 3
o5 = total: 1 5 8 4
0: 1 . . .
1: . . . .
2: . 2 . .
3: . 3 2 .
4: . . 4 2
5: . . 2 2
o5 : BettiTally</pre>
</td></tr>
</table>
Note that the entries are displayed in a peculiar manner. Let's lift this to the integers.<table class="examples"><tr><td><pre>i6 : lift(value C, ZZ)
0 1 2 3
o6 = total: 1 5 8 4
0: 1 . . .
1: . . . .
2: . 2 . .
3: . 3 2 .
4: . . 4 2
5: . . 2 2
o6 : BettiTally</pre>
</td></tr>
<tr><td><pre>i7 : B == oo
o7 = true</pre>
</td></tr>
</table>
Let's display the list of Betti diagrams in the decomposition, and also the list of multipliers.<table class="examples"><tr><td><pre>i8 : netList pack(3, apply(toList C, x -> x#1))
+-----------------+-----------------+---------------+
| 0 1 2 3| 0 1 2 3 | 0 1 2 3|
o8 = |total: 8 35 42 15|total: 2 7 14 9 |total: 1 7 14 8|
| 0: 8 . . .| 0: 2 . . . | 0: 1 . . .|
| 1: . . . .| 1: . . . . | 1: . . . .|
| 2: . 35 . .| 2: . 7 . . | 2: . . . .|
| 3: . . 42 .| 3: . . . . | 3: . 7 . .|
| 4: . . . 15| 4: . . 14 9 | 4: . . 14 8|
+-----------------+-----------------+---------------+
| 0 1 2 3 | 0 1 2 3| |
|total: 1 6 8 3 |total: 3 14 32 21| |
| 0: 1 . . . | 0: 3 . . .| |
| 1: . . . . | 1: . . . .| |
| 2: . . . . | 2: . . . .| |
| 3: . 6 . . | 3: . 14 . .| |
| 4: . . 8 . | 4: . . . .| |
| 5: . . . 3 | 5: . . 32 21| |
+-----------------+-----------------+---------------+</pre>
</td></tr>
<tr><td><pre>i9 : apply(toList C, first)
1 1 3 11 1
o9 = {--, --, --, --, --}
21 21 28 48 16
o9 : List</pre>
</td></tr>
</table>
Here is an example where the Betti diagram is not Cohen-Macaulay.<table class="examples"><tr><td><pre>i10 : R = ZZ/103[a,b,c]
o10 = R
o10 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i11 : I = ideal"a3,abc,b4,b2c2"
3 4 2 2
o11 = ideal (a , a*b*c, b , b c )
o11 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i12 : B = betti res I
0 1 2 3
o12 = total: 1 4 5 2
0: 1 . . .
1: . . . .
2: . 2 . .
3: . 2 2 .
4: . . 2 1
5: . . 1 1
o12 : BettiTally</pre>
</td></tr>
<tr><td><pre>i13 : C = decompose B
1 / 0 1 2 3\ 2 / 0 1 2 3\ 1 / 0 1 2 3\ 17 / 0 1 2 3\ 13 / 0 1 2 3\ 2 / 0 1 2\
o13 = (--)|total: 8 35 42 15| + (--)|total: 2 7 14 9| + (---)|total: 5 16 20 9| + (--)|total: 1 6 8 3| + (---)|total: 3 14 32 21| + (--)|total: 3 7 4|
21 | 0: 8 . . .| 63 | 0: 2 . . .| 144 | 0: 5 . . .| 96 | 0: 1 . . .| 672 | 0: 3 . . .| 21 | 0: 3 . .|
| 1: . . . .| | 1: . . . .| | 1: . . . .| | 1: . . . .| | 1: . . . .| | 1: . . .|
| 2: . 35 . .| | 2: . 7 . .| | 2: . 16 . .| | 2: . . . .| | 2: . . . .| | 2: . . .|
| 3: . . 42 .| | 3: . . . .| | 3: . . . .| | 3: . 6 . .| | 3: . 14 . .| | 3: . 7 .|
\ 4: . . . 15/ \ 4: . . 14 9/ | 4: . . 20 .| | 4: . . 8 .| | 4: . . . .| | 4: . . .|
\ 5: . . . 9/ \ 5: . . . 3/ \ 5: . . 32 21/ \ 5: . . 4/
o13 : Expression of class Sum</pre>
</td></tr>
</table>
The following example cannot be decomposed. This means that there is no module with this Betti diagram.<table class="examples"><tr><td><pre>i14 : M = matrix"1,0,0,0;0,0,0,0;0,3,0,0;0,0,5,3"
o14 = | 1 0 0 0 |
| 0 0 0 0 |
| 0 3 0 0 |
| 0 0 5 3 |
4 4
o14 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i15 : B = mat2betti M
0 1 2 3
o15 = total: 1 3 5 3
0: 1 . . .
1: . . . .
2: . 3 . .
3: . . 5 3
o15 : BettiTally</pre>
</td></tr>
<tr><td><pre>i16 : codim B
o16 = 1</pre>
</td></tr>
<tr><td><pre>i17 : degree B
o17 = 2</pre>
</td></tr>
<tr><td><pre>i18 : try decompose B else "Betti diagram cannot exist"
NOT IN THIS SIMPLEX OF PURE BETTI DIAGRAMS
o18 = Betti diagram cannot exist</pre>
</td></tr>
<tr><td><pre>i19 : pureBettiDiagram lowestDegrees B
0 1 2 3
o19 = total: 1 5 9 5
0: 1 . . .
1: . . . .
2: . 5 . .
3: . . 9 5
o19 : BettiTally</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_pure__Betti__Diagram_lp__List_rp.html" title="pure Betti diagram given a list of degrees">pureBettiDiagram</a> -- pure Betti diagram given a list of degrees</span></li>
<li><span><a href="../../Macaulay2Doc/html/_betti.html" title="display degrees">betti</a> -- display degrees</span></li>
<li><span><a href="../../Macaulay2Doc/html/_value.html" title="evaluate">value</a> -- evaluate</span></li>
<li><span><a href="../../Macaulay2Doc/html/_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></li>
<li><span><a href="../../Macaulay2Doc/html/_to__List.html" title="list of elements">toList</a> -- list of elements</span></li>
<li><span><a href="../../Macaulay2Doc/html/_pack.html" title="pack elements of a list into shorter ones">pack</a> -- pack elements of a list into shorter ones</span></li>
</ul>
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