By default the option FineGrading is set to false. With FineGrading=>true the script returns the ZZ4-graded resolution, and the function h returns the homotopies one graded component at a time as a HashTable.
Note that the homotopies are 0 except in the middle part of the resolution, where there is a generator degree common to two consecutive free modules.
i1 : (F,h0) = canonicalHomotopies(7,3)
ZZ 1 ZZ
o1 = ((-----[x , x , x , x , y , y , y , y ]) <-- (-----[x , x , x , x , y ,
32003 0 1 2 3 0 1 2 3 32003 0 1 2 3 0
0 1
------------------------------------------------------------------------
10 ZZ 16
y , y , y ]) <-- (-----[x , x , x , x , y , y , y , y ]) <--
1 2 3 32003 0 1 2 3 0 1 2 3
2
------------------------------------------------------------------------
ZZ 16 ZZ
(-----[x , x , x , x , y , y , y , y ]) <-- (-----[x , x , x , x , y ,
32003 0 1 2 3 0 1 2 3 32003 0 1 2 3 0
3 4
------------------------------------------------------------------------
10 ZZ 1
y , y , y ]) <-- (-----[x , x , x , x , y , y , y , y ]) <-- 0, h0)
1 2 3 32003 0 1 2 3 0 1 2 3
6
5
o1 : Sequence
|
i2 : betti F
0 1 2 3 4 5
o2 = total: 1 10 16 16 10 1
0: 1 . . . . .
1: . 10 16 . . .
2: . . . 16 10 .
3: . . . . . 1
o2 : BettiTally
|
i3 : netList apply(length F, j-> sum(rank F_1, i->h0(i,j)))
+-------------------------------------------------------+
o3 = |{2} | 1 | |
|{2} | 1 | |
|{2} | 1 | |
|{2} | 1 | |
|{2} | 1 | |
|{2} | 1 | |
|{2} | 1 | |
|{2} | 1 | |
|{2} | 1 | |
|{2} | 1 | |
+-------------------------------------------------------+
|0 |
+-------------------------------------------------------+
|{5} | -1 -1 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 ||
|{5} | -1 -1 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 ||
|{5} | -1 0 -2 0 0 2 0 0 -2 -2 -1 0 0 0 0 0 ||
|{5} | 0 1 -2 0 0 2 0 0 0 -2 0 -1 0 0 0 0 ||
|{5} | -1 0 0 -1 0 0 1 0 -1 0 0 0 -1 0 0 0 ||
|{5} | 0 1 0 -1 0 0 1 0 0 -1 0 0 0 -1 0 0 ||
|{5} | 0 0 2 -1 0 0 0 0 2 -2 0 0 0 0 -1 0 ||
|{5} | 1 0 0 0 -2 0 0 2 0 -2 -1 0 2 0 0 0 ||
|{5} | 0 1 0 0 2 0 0 -2 0 0 0 1 0 -2 0 0 ||
|{5} | 0 0 1 0 1 0 0 0 0 -2 -1 1 0 0 -1 0 ||
|{5} | 0 0 0 1 2 0 0 0 0 0 0 0 -2 2 -1 0 ||
|{5} | 0 0 0 0 0 -2 1 0 -2 2 0 0 0 0 0 -1 ||
|{5} | 0 0 0 0 0 -1 0 -1 0 2 1 -1 0 0 0 -1 ||
|{5} | 0 0 0 0 0 0 -1 -2 0 0 0 0 2 -2 0 -1 ||
|{5} | 0 0 0 0 0 0 0 0 -1 -2 -1 0 1 0 -1 -1 ||
|{5} | 0 0 0 0 0 0 0 0 0 -1 0 -1 0 1 -1 -1 ||
+-------------------------------------------------------+
|0 |
+-------------------------------------------------------+
|{8} | -1 -1 1 -1 1 1 1 -1 1 1 | |
+-------------------------------------------------------+
|
i4 : H = makeHomotopies1(F.dd_1, F); |
i5 : (F,h0) = canonicalHomotopies(7,3, FineGrading=>true); |
i6 : h0(0,2)
o6 = HashTable{{0, 5, 6, 9} => 0 }
{0, 5, 7, 8} => 0
{1, 4, 5, 10} => 0
{1, 4, 6, 9} => 0
{1, 4, 7, 8} => {1, 4, 7, 8} | 0 1 |
{1, 4, 8, 7} => {1, 4, 8, 7} | 1 |
{2, 3, 5, 10} => 0
{2, 3, 6, 9} => {2, 3, 6, 9} | -1 0 |
{2, 3, 7, 8} => {2, 3, 7, 8} | 1 0 |
{2, 3, 7, 8} | -2 1 |
{2, 3, 8, 7} => {2, 3, 8, 7} | 0 |
{2, 3, 8, 7} | 1 |
{2, 3, 9, 6} => 0
{3, 2, 6, 9} => {3, 2, 6, 9} | -1 |
{3, 2, 7, 8} => {3, 2, 7, 8} | 1 |
{3, 2, 7, 8} | 0 |
{3, 2, 8, 7} => 0
{3, 2, 9, 6} => 0
{4, 1, 7, 8} => 0
{4, 1, 8, 7} => 0
o6 : HashTable
|
i7 : homotopyRanks(7,3)
0 1 2 3 4 5
total: 1 10 16 16 10 1
0: 1 . . . . .
1: . 10 16 . . .
2: . . . 16 10 .
3: . . . . . 1
+-------------------------+---------------+
o7 = || x_1^2-x_0x_2 | |{1, 0, 8, 0, 1}|
+-------------------------+---------------+
|| x_1x_2-x_0x_3 | |{1, 0, 8, 0, 1}|
+-------------------------+---------------+
|| x_2^2-x_1x_3 | |{1, 0, 8, 0, 1}|
+-------------------------+---------------+
|| x_2y_0-2x_1y_1+x_0y_2 ||{1, 0, 8, 0, 1}|
+-------------------------+---------------+
|| x_3y_0-2x_2y_1+x_1y_2 ||{1, 0, 8, 0, 1}|
+-------------------------+---------------+
|| x_2y_1-2x_1y_2+x_0y_3 ||{1, 0, 8, 0, 1}|
+-------------------------+---------------+
|| x_3y_1-2x_2y_2+x_1y_3 ||{1, 0, 8, 0, 1}|
+-------------------------+---------------+
|| y_1^2-y_0y_2 | |{1, 0, 8, 0, 1}|
+-------------------------+---------------+
|| y_1y_2-y_0y_3 | |{1, 0, 8, 0, 1}|
+-------------------------+---------------+
|| y_2^2-y_1y_3 | |{1, 0, 8, 0, 1}|
+-------------------------+---------------+
|