of this matrix will be >= 0 for every minimal free resolution. Of course, the converse does not hold!
i1 : d = {0,2,3,6,7,9}
o1 = {0, 2, 3, 6, 7, 9}
o1 : List
|
i2 : de = {0,2,4,6,7,9}
o2 = {0, 2, 4, 6, 7, 9}
o2 : List
|
i3 : e = {0,3,4,6,7,9}
o3 = {0, 3, 4, 6, 7, 9}
o3 : List
|
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
|
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
|
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
|
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
|
i8 : dotProduct(C,B1)
o8 = 0
|
i9 : dotProduct(C,B2)
o9 = 945
|
i10 : dotProduct(C,B3)
o10 = 0
|
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.
i11 : d = {-4,-3,0,2,3,6,7,9}
o11 = {-4, -3, 0, 2, 3, 6, 7, 9}
o11 : List
|
i12 : de = {-4,-3,0,2,4,6,7,9}
o12 = {-4, -3, 0, 2, 4, 6, 7, 9}
o12 : List
|
i13 : e = {-4,-3,0,3,4,6,7,9}
o13 = {-4, -3, 0, 3, 4, 6, 7, 9}
o13 : List
|
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
|
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
|
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
|
Let's check that this is zero on the appropriate pure diagrams, and positive on the one corresponding to de: