<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>facetEquation(List,ZZ,ZZ,ZZ) -- The upper facet equation corresponding to (L,i)</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_highest__Degrees_lp__Betti__Tally_rp.html">next</a> | <a href="_dot__Product.html">previous</a> | <a href="_highest__Degrees_lp__Betti__Tally_rp.html">forward</a> | <a href="_dot__Product.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>facetEquation(List,ZZ,ZZ,ZZ) -- The upper facet equation corresponding to (L,i)</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>facetEquation(L,i,lodeg,hideg)</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_facet__Equation_lp__List_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.html" title="The upper facet equation corresponding to (L,i)">facetEquation</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, the degree sequence, a strictly ascending list of integers</span></li>
<li><span><tt>i</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, an index in range 0..#L-2, such that L_(i+1) == L_i + 2</span></li>
<li><span><tt>lodeg</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the leftmost degree of the table</span></li>
<li><span><tt>hideg</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the rightmost degree of the table</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span>A (hideg-lodeg+1) by #L matrix over ZZ whose rows correspond to slanted degrees lodeg .. hideg, such that the dot product of this matrix with any betti diagram of any finite length module is >= 0.</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>The (entry by entry) <a href="_dot__Product.html">dot product</a> of this matrix will be >= 0 for every minimal free resolution. Of course, the converse does not hold!<table class="examples"><tr><td><pre>i1 : d = {0,2,3,6,7,9}
o1 = {0, 2, 3, 6, 7, 9}
o1 : List</pre>
</td></tr>
<tr><td><pre>i2 : de = {0,2,4,6,7,9}
o2 = {0, 2, 4, 6, 7, 9}
o2 : List</pre>
</td></tr>
<tr><td><pre>i3 : e = {0,3,4,6,7,9}
o3 = {0, 3, 4, 6, 7, 9}
o3 : List</pre>
</td></tr>
<tr><td><pre>i4 : B1 = pureBettiDiagram d
0 1 2 3 4 5
o4 = total: 10 81 105 105 81 10
0: 10 . . . . .
1: . 81 105 . . .
2: . . . . . .
3: . . . 105 81 .
4: . . . . . 10
o4 : BettiTally</pre>
</td></tr>
<tr><td><pre>i5 : B2 = pureBettiDiagram de
0 1 2 3 4 5
o5 = total: 5 27 63 105 72 8
0: 5 . . . . .
1: . 27 . . . .
2: . . 63 . . .
3: . . . 105 72 .
4: . . . . . 8
o5 : BettiTally</pre>
</td></tr>
<tr><td><pre>i6 : B3 = pureBettiDiagram e
0 1 2 3 4 5
o6 = total: 5 105 189 210 135 14
0: 5 . . . . .
1: . . . . . .
2: . 105 189 . . .
3: . . . 210 135 .
4: . . . . . 14
o6 : BettiTally</pre>
</td></tr>
<tr><td><pre>i7 : C = facetEquation(de,1,0,6)
o7 = | 0 30 -35 27 -15 5 |
| 0 35 -27 15 -5 0 |
| 0 0 0 5 0 0 |
| 0 0 0 0 0 2 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
7 6
o7 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i8 : dotProduct(C,B1)
o8 = 0</pre>
</td></tr>
<tr><td><pre>i9 : dotProduct(C,B2)
o9 = 945</pre>
</td></tr>
<tr><td><pre>i10 : dotProduct(C,B3)
o10 = 0</pre>
</td></tr>
</table>
The following example is from Eisenbud and Schreyer, math.AC/0712.1843v2, example 2.4. Notice that the notation here differs slightly from theirs. In both cases i refers to the index, but in Macaulay2, the first element of a list has index 0. hence in this example, i is 3 and not 4, as in the example in the paper.<table class="examples"><tr><td><pre>i11 : d = {-4,-3,0,2,3,6,7,9}
o11 = {-4, -3, 0, 2, 3, 6, 7, 9}
o11 : List</pre>
</td></tr>
<tr><td><pre>i12 : de = {-4,-3,0,2,4,6,7,9}
o12 = {-4, -3, 0, 2, 4, 6, 7, 9}
o12 : List</pre>
</td></tr>
<tr><td><pre>i13 : e = {-4,-3,0,3,4,6,7,9}
o13 = {-4, -3, 0, 3, 4, 6, 7, 9}
o13 : List</pre>
</td></tr>
<tr><td><pre>i14 : pureBettiDiagram d
0 1 2 3 4 5 6 7
o14 = total: 405 1001 3575 11583 10725 5005 3159 275
-4: 405 1001 . . . . . .
-3: . . . . . . . .
-2: . . 3575 . . . . .
-1: . . . 11583 10725 . . .
0: . . . . . . . .
1: . . . . . 5005 3159 .
2: . . . . . . . 275
o14 : BettiTally</pre>
</td></tr>
<tr><td><pre>i15 : pureBettiDiagram de
0 1 2 3 4 5 6 7
o15 = total: 945 2288 7150 15444 19305 20020 11232 880
-4: 945 2288 . . . . . .
-3: . . . . . . . .
-2: . . 7150 . . . . .
-1: . . . 15444 . . . .
0: . . . . 19305 . . .
1: . . . . . 20020 11232 .
2: . . . . . . . 880
o15 : BettiTally</pre>
</td></tr>
<tr><td><pre>i16 : C = facetEquation(de,3,-6,3)
o16 = | 1755 -385 0 0 66 -70 0 100 |
| 385 0 0 -66 70 0 -100 175 |
| 0 0 66 -70 0 100 -175 189 |
| 0 0 70 0 -100 175 -189 140 |
| 0 0 0 100 -175 189 -140 60 |
| 0 0 0 175 -189 140 -60 0 |
| 0 0 0 0 0 60 0 0 |
| 0 0 0 0 0 0 0 44 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
10 8
o16 : Matrix ZZ <--- ZZ</pre>
</td></tr>
</table>
Let's check that this is zero on the appropriate pure diagrams, and positive on the one corresponding to de:<table class="examples"><tr><td><pre>i17 : dotProduct(C,-6,pureBettiDiagram d)
o17 = 0</pre>
</td></tr>
<tr><td><pre>i18 : dotProduct(C,-6,pureBettiDiagram de)
o18 = 2702700</pre>
</td></tr>
<tr><td><pre>i19 : dotProduct(C,-6,pureBettiDiagram e)
o19 = 0</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_dot__Product.html" title="entry by entry dot product of two Betti diagrams">dotProduct</a> -- entry by entry dot product of two Betti diagrams</span></li>
<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>
</ul>
</div>
</div>
</body>
</html>
